let A be Euclidean preIfWhileAlgebra; :: thesis: for X being non empty countable set
for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let X be non empty countable set ; :: thesis: for s being Element of Funcs (X,INT)
for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let s be Element of Funcs (X,INT); :: thesis: for T being Subset of (Funcs (X,INT))
for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); :: thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; :: thesis: for x being Variable of f
for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let x be Variable of f; :: thesis: for t being INT-Expression of A,f holds
( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
let t be INT-Expression of A,f; :: thesis: ( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) )
A1: (^ x) . s = x ;
dom ((. x) + t) = Funcs (X,INT) by FUNCT_2:def 1;
then A2: ((. x) + t) . s = ((. x) . s) + (t . s) by VALUED_1:def 1;
(. x) . s = s . x by Th22;
hence ( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds
(f . (s,(x += t))) . z = s . z ) ) by A1, A2, Th24; :: thesis: verum