let S be non empty non void bool-correct 4,1 integer BoolSignature ; :: thesis: for X being non-empty ManySortedSet of the carrier of S
for T being b₁,S -terms all_vars_including inheriting_operations free_in_itself vf-free integer VarMSAlgebra over S
for C being bool-correct 4,1 integer image of T
for I being integer SortSymbol of S
for i, j being Integer
for a, b being Element of C,I st a = i & b = j holds
a - b = i - j
let X be non-empty ManySortedSet of the carrier of S; :: thesis: for T being X,S -terms all_vars_including inheriting_operations free_in_itself vf-free integer VarMSAlgebra over S
for C being bool-correct 4,1 integer image of T
for I being integer SortSymbol of S
for i, j being Integer
for a, b being Element of C,I st a = i & b = j holds
a - b = i - j
let T be X,S -terms all_vars_including inheriting_operations free_in_itself vf-free integer VarMSAlgebra over S; :: thesis: for C being bool-correct 4,1 integer image of T
for I being integer SortSymbol of S
for i, j being Integer
for a, b being Element of C,I st a = i & b = j holds
a - b = i - j
let C be bool-correct 4,1 integer image of T; :: thesis: for I being integer SortSymbol of S
for i, j being Integer
for a, b being Element of C,I st a = i & b = j holds
a - b = i - j
let I be integer SortSymbol of S; :: thesis: for i, j being Integer
for a, b being Element of C,I st a = i & b = j holds
a - b = i - j
let i, j be Integer; :: thesis: for a, b being Element of C,I st a = i & b = j holds
a - b = i - j
let a, b be Element of C,I; :: thesis: ( a = i & b = j implies a - b = i - j )
assume A1: a = i ; :: thesis: ( not b = j or a - b = i - j )
assume b = j ; :: thesis: a - b = i - j
then - b = - j by AOFA_A00:55;
then a + (- b) = i + (- j) by A1, AOFA_A00:55;
hence a - b = i - j ; :: thesis: verum