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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnltlem1 8101 Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
 M  NN  N  NN  M  <  N  M  <_  N  -  1
 
Theoremnnm1ge0 8102 A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018.)
 N  NN  0  <_  N  -  1
 
Theoremnn0ge0div 8103 Division of a nonnegative integer by a positive number is not negative. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
 K  NN0  L  NN  0  <_  K  L
 
Theoremzdiv 8104* Two ways to express " M divides  N. (Contributed by NM, 3-Oct-2008.)
 M  NN  N  ZZ  k  ZZ  M  x.  k  N  N  M  ZZ
 
Theoremzdivadd 8105 Property of divisibility: if  D divides and then it divides  + . (Contributed by NM, 3-Oct-2008.)
 D  NN  ZZ  ZZ  D  ZZ  D 
 ZZ  +  D 
 ZZ
 
Theoremzdivmul 8106 Property of divisibility: if  D divides then it divides  x. . (Contributed by NM, 3-Oct-2008.)
 D  NN  ZZ  ZZ  D  ZZ  x.  D  ZZ
 
Theoremzextle 8107* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
 M  ZZ  N  ZZ  k  ZZ  k  <_  M  k  <_  N  M  N
 
Theoremzextlt 8108* An extensionality-like property for integer ordering. (Contributed by NM, 29-Oct-2005.)
 M  ZZ  N  ZZ  k  ZZ  k  <  M  k  <  N  M  N
 
Theoremrecnz 8109 The reciprocal of a number greater than 1 is not an integer. (Contributed by NM, 3-May-2005.)
 RR  1  <  1  ZZ
 
Theorembtwnnz 8110 A number between an integer and its successor is not an integer. (Contributed by NM, 3-May-2005.)
 ZZ  <  <  +  1  ZZ
 
Theoremgtndiv 8111 A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)
 RR  NN  <  ZZ
 
Theoremhalfnz 8112 One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
 1  2  ZZ
 
Theoremprime 8113* Two ways to express " is a prime number (or 1)." (Contributed by NM, 4-May-2005.)
 NN  NN  NN  1  NN  1  <  <_  NN
 
Theoremmsqznn 8114 The square of a nonzero integer is a positive integer. (Contributed by NM, 2-Aug-2004.)
 ZZ  =/=  0  x.  NN
 
Theoremzneo 8115 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
 ZZ  ZZ  2  x.  =/=  2  x.  +  1
 
Theoremnneoor 8116 A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)
 N  NN  N  2  NN  N  +  1  2  NN
 
Theoremnneo 8117 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
 N  NN  N  2  NN  N  +  1  2  NN
 
Theoremnneoi 8118 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.)
 N  NN   =>     N  2  NN  N  +  1  2  NN
 
Theoremzeo 8119 An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
 N  ZZ  N  2  ZZ  N  +  1  2  ZZ
 
Theoremzeo2 8120 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.)
 N  ZZ  N  2  ZZ  N  +  1  2  ZZ
 
Theorempeano2uz2 8121* Second Peano postulate for upper integers. (Contributed by NM, 3-Oct-2004.)
 ZZ  {  ZZ  |  <_  }  +  1  {  ZZ  |  <_  }
 
Theorempeano5uzti 8122* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
 N  ZZ  N  +  1  { k  ZZ  |  N  <_  k }  C_
 
Theorempeano5uzi 8123* Peano's inductive postulate for upper integers. (Contributed by NM, 6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
 N  ZZ   =>     N  +  1  { k  ZZ  |  N  <_  k }  C_
 
Theoremdfuzi 8124* An expression for the upper integers that start at  N that is analogous to dfnn2 7697 for positive integers. (Contributed by NM, 6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
 N  ZZ   =>     {  ZZ  |  N  <_  }  |^| {  |  N  +  1  }
 
Theoremuzind 8125* Induction on the upper integers that start at  M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 5-Jul-2005.)
 j  M    &     j  k    &     j  k  +  1    &     j  N    &     M  ZZ    &     M  ZZ  k  ZZ  M  <_  k    =>     M  ZZ  N  ZZ  M  <_  N
 
Theoremuzind2 8126* Induction on the upper integers that start after an integer  M. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.)
 j  M  +  1    &     j  k    &     j  k  +  1    &     j  N    &     M  ZZ    &     M  ZZ  k  ZZ  M  <  k    =>     M  ZZ  N  ZZ  M  <  N
 
Theoremuzind3 8127* Induction on the upper integers that start at an integer  M. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 26-Jul-2005.)
 j  M    &     j  m    &     j  m  +  1    &     j  N    &     M  ZZ    &     M  ZZ  m  {
 k  ZZ  |  M  <_  k }    =>     M  ZZ  N  {
 k  ZZ  |  M  <_  k }
 
Theoremnn0ind 8128* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 13-May-2004.)
 0    &       &     +  1    &       &       &     NN0    =>     NN0
 
Theoremfzind 8129* Induction on the integers from  M to  N inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
 M    &       &     +  1    &     K    &     M  ZZ  N  ZZ  M  <_  N    &     M  ZZ  N  ZZ  ZZ  M  <_  <  N    =>     M  ZZ  N  ZZ  K  ZZ  M  <_  K  K  <_  N
 
Theoremfnn0ind 8130* Induction on the integers from  0 to  N inclusive . The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by Paul Chapman, 31-Mar-2011.)
 0    &       &     +  1    &     K    &     N  NN0    &     N  NN0  NN0  <  N    =>     N  NN0  K  NN0  K  <_  N
 
Theoremnn0ind-raph 8131* Principle of Mathematical Induction (inference schema) on nonnegative integers. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. Raph Levien remarks: "This seems a bit painful. I wonder if an explicit substitution version would be easier." (Contributed by Raph Levien, 10-Apr-2004.)
 0    &       &     +  1    &       &       &     NN0    =>     NN0
 
Theoremzindd 8132* Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.)
 0    &       &     +  1    &     -u    &       &       &     NN0    &     NN    =>     ZZ
 
Theorembtwnz 8133* Any real number can be sandwiched between two integers. Exercise 2 of [Apostol] p. 28. (Contributed by NM, 10-Nov-2004.)
 RR  ZZ  <  ZZ  <
 
Theoremnn0zd 8134 A positive integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 NN0   =>     ZZ
 
Theoremnnzd 8135 A nonnegative integer is an integer. (Contributed by Mario Carneiro, 28-May-2016.)
 NN   =>     ZZ
 
Theoremzred 8136 An integer is a real number. (Contributed by Mario Carneiro, 28-May-2016.)
 ZZ   =>     RR
 
Theoremzcnd 8137 An integer is a complex number. (Contributed by Mario Carneiro, 28-May-2016.)
 ZZ   =>     CC
 
Theoremznegcld 8138 Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.)
 ZZ   =>     -u  ZZ
 
Theorempeano2zd 8139 Deduction from second Peano postulate generalized to integers. (Contributed by Mario Carneiro, 28-May-2016.)
 ZZ   =>     +  1 
 ZZ
 
Theoremzaddcld 8140 Closure of addition of integers. (Contributed by Mario Carneiro, 28-May-2016.)
 ZZ   &     ZZ   =>     + 
 ZZ
 
Theoremzsubcld 8141 Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016.)
 ZZ   &     ZZ   =>     - 
 ZZ
 
Theoremzmulcld 8142 Closure of multiplication of integers. (Contributed by Mario Carneiro, 28-May-2016.)
 ZZ   &     ZZ   =>     x. 
 ZZ
 
Theoremzadd2cl 8143 Increasing an integer by 2 results in an integer. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
 N  ZZ  N  +  2  ZZ
 
3.4.9  Decimal arithmetic
 
Syntaxcdc 8144 Constant used for decimal constructor.
;
 
Definitiondf-dec 8145 Define the "decimal constructor", which is used to build up "decimal integers" or "numeric terms" in base 10. For example, ;;; 1 0 0 0  + ;;; 2 0 0 0 ;;; 3 0 0 0. (Contributed by Mario Carneiro, 17-Apr-2015.)
;  10  x.  +
 
Theoremdeceq1 8146 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; C ; C
 
Theoremdeceq2 8147 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
; C ; C
 
Theoremdeceq1i 8148 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
   =>    ; C ; C
 
Theoremdeceq2i 8149 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
   =>    ; C ; C
 
Theoremdeceq12i 8150 Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.)
   &     C  D   =>    ; C ; D
 
Theoremnumnncl 8151 Closure for a numeral (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN   =>     T  x.  +  NN
 
Theoremnum0u 8152 Add a zero in the units place. (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   =>     T  x.  T  x.  +  0
 
Theoremnum0h 8153 Add a zero in the higher places. (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   =>     T  x.  0  +
 
Theoremnumcl 8154 Closure for a decimal integer (with units place). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   =>     T  x.  +  NN0
 
Theoremnumsuc 8155 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   &     +  1  C   &     N  T  x.  +    =>     N  +  1  T  x.  +  C
 
Theoremdecnncl 8156 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     NN   =>    ;  NN
 
Theoremdeccl 8157 Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     NN0   =>    ;  NN0
 
Theoremdec0u 8158 Add a zero in the units place. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   =>     10  x. ; 0
 
Theoremdec0h 8159 Add a zero in the higher places. (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   =>    ; 0
 
Theoremnumnncl2 8160 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 9-Mar-2015.)
 T  NN   &     NN   =>     T  x.  +  0  NN
 
Theoremdecnncl2 8161 Closure for a decimal integer (zero units place). (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN   =>    ; 0  NN
 
Theoremnumlt 8162 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN   &     NN0   &     NN0   &     C  NN   &     <  C   =>     T  x.  +  <  T  x.  +  C
 
Theoremnumltc 8163 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     C  <  T   &     <    =>     T  x.  +  C  <  T  x.  +  D
 
Theoremdeclt 8164 Comparing two decimal integers (equal higher places). (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     NN0   &     C  NN   &     <  C   =>    ;  < ; C
 
Theoremdecltc 8165 Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     C  <  10   &     <    =>    ; C  < ; D
 
Theoremdecsuc 8166 The successor of a decimal integer (no carry). (Contributed by Mario Carneiro, 17-Apr-2015.)
 NN0   &     NN0   &     +  1  C   &     N ;   =>     N  +  1 ; C
 
Theoremnumlti 8167 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN   &     NN   &     NN0   &     C  NN0   &     C  <  T   =>     C  <  T  x.  +
 
Theoremdeclti 8168 Comparing a digit to a decimal integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN   &     NN0   &     C  NN0   &     C  <  10   =>     C  < ;
 
Theoremnumsucc 8169 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 Y  NN0   &     T  Y  +  1   &     NN0   &     +  1    &     N  T  x.  +  Y   =>     N  +  1  T  x.  +  0
 
Theoremdecsucc 8170 The successor of a decimal integer (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     +  1    &     N ; 9   =>     N  +  1 ; 0
 
Theorem1e0p1 8171 The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
 1  0  +  1
 
Theoremdec10p 8172 Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 10  + ; 1
 
Theoremdec10 8173 The decimal form of 10. NB: In our presentations of large numbers later on, we will use our symbol for 10 at the highest digits when advantageous, because we can use this theorem to convert back to "long form" (where each digit is in the range 0-9) with no extra effort. However, we cannot do this for lower digits while maintaining the ease of use of the decimal system, since it requires nontrivial number knowledge (more than just equality theorems) to convert back. (Contributed by Mario Carneiro, 18-Feb-2014.)

 10 ; 1 0
 
Theoremnumma 8174 Perform a multiply-add of two decimal integers  M and 
N against a fixed multiplicand  P (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     P  NN0   &     x.  P  +  C  E   &     x.  P  +  D  F   =>     M  x.  P  +  N  T  x.  E  +  F
 
Theoremnummac 8175 Perform a multiply-add of two decimal integers  M and 
N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     P  NN0   &     F  NN0   &     G  NN0   &     x.  P  +  C  +  G  E   &     x.  P  +  D  T  x.  G  +  F   =>     M  x.  P  +  N  T  x.  E  +  F
 
Theoremnumma2c 8176 Perform a multiply-add of two decimal integers  M and 
N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     P  NN0   &     F  NN0   &     G  NN0   &     P  x.  +  C  +  G  E   &     P  x.  +  D  T  x.  G  +  F   =>     P  x.  M  +  N  T  x.  E  +  F
 
Theoremnumadd 8177 Add two decimal integers  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     +  C  E   &     +  D  F   =>     M  +  N  T  x.  E  +  F
 
Theoremnumaddc 8178 Add two decimal integers  M and  N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     NN0   &     NN0   &     C  NN0   &     D  NN0   &     M  T  x.  +    &     N  T  x.  C  +  D   &     F  NN0   &     +  C  +  1  E   &     +  D  T  x.  1  +  F   =>     M  +  N  T  x.  E  +  F
 
Theoremnummul1c 8179 The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     P  NN0   &     NN0   &     NN0   &     N  T  x.  +    &     D  NN0   &     E  NN0   &     x.  P  +  E  C   &     x.  P  T  x.  E  +  D   =>     N  x.  P  T  x.  C  +  D
 
Theoremnummul2c 8180 The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 T  NN0   &     P  NN0   &     NN0   &     NN0   &     N  T  x.  +    &     D  NN0   &     E  NN0   &     P  x.  +  E  C   &     P  x.  T  x.  E  +  D   =>     P  x.  N  T  x.  C  +  D
 
Theoremdecma 8181 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     P  NN0   &     x.  P  +  C  E   &     x.  P  +  D  F   =>     M  x.  P  +  N ; E F
 
Theoremdecmac 8182 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     P  NN0   &     F  NN0   &     G  NN0   &     x.  P  +  C  +  G  E   &     x.  P  +  D ; G F   =>     M  x.  P  +  N ; E F
 
Theoremdecma2c 8183 Perform a multiply-add of two numerals  M and  N against a fixed multiplicand  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     P  NN0   &     F  NN0   &     G  NN0   &     P  x.  +  C  +  G  E   &     P  x.  +  D ; G F   =>     P  x.  M  +  N ; E F
 
Theoremdecadd 8184 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     +  C  E   &     +  D  F   =>     M  +  N ; E F
 
Theoremdecaddc 8185 Add two numerals  M and  N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     +  C  +  1  E   &     F  NN0   &     +  D ; 1 F   =>     M  +  N ; E F
 
Theoremdecaddc2 8186 Add two numerals  M and  N (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     C  NN0   &     D  NN0   &     M ;   &     N ; C D   &     +  C  +  1  E   &     +  D  10   =>     M  +  N ; E 0
 
Theoremdecaddi 8187 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     N  NN0   &     M ;   &     +  N  C   =>     M  +  N ; C
 
Theoremdecaddci 8188 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     N  NN0   &     M ;   &     +  1  D   &     C  NN0   &     +  N ; 1 C   =>     M  +  N ; D C
 
Theoremdecaddci2 8189 Add two numerals  M and  N (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 NN0   &     NN0   &     N  NN0   &     M ;   &     +  1  D   &     +  N  10   =>     M  +  N ; D 0
 
Theoremdecmul1c 8190 The product of a numeral with a number. (Contributed by Mario Carneiro, 18-Feb-2014.)
 P  NN0   &     NN0   &     NN0   &     N ;   &     D  NN0   &     E  NN0   &     x.  P  +  E  C   &     x.  P ; E D   =>     N  x.  P ; C D
 
Theoremdecmul2c 8191 The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
 P  NN0   &     NN0   &     NN0   &     N ;   &     D  NN0   &     E  NN0   &     P  x.  +  E  C   &     P  x. ; E D   =>     P  x.  N ; C D
 
Theorem6p5lem 8192 Lemma for 6p5e11 8193 and related theorems. (Contributed by Mario Carneiro, 19-Apr-2015.)
 NN0   &     D  NN0   &     E  NN0   &     D  +  1   &     C  E  +  1   &     +  D ; 1 E   =>     + ; 1 C
 
Theorem6p5e11 8193 6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  +  5 ; 1 1
 
Theorem6p6e12 8194 6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 6  +  6 ; 1 2
 
Theorem7p4e11 8195 7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  +  4 ; 1 1
 
Theorem7p5e12 8196 7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  +  5 ; 1 2
 
Theorem7p6e13 8197 7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  +  6 ; 1 3
 
Theorem7p7e14 8198 7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
 7  +  7 ; 1 4
 
Theorem8p3e11 8199 8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  3 ; 1 1
 
Theorem8p4e12 8200 8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
 8  +  4 ; 1 2
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