let A be Euclidean preIfWhileAlgebra; 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 ; 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); 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)); 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; 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; 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; ( (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 4;
(. 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; verum