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 i being Integer
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . 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 i being Integer
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . 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 i being Integer
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let T be Subset of (Funcs (X,INT)); :: thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T
for i being Integer
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; :: thesis: for i being Integer
for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let i be Integer; :: thesis: for x being Variable of f holds
( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
let x be Variable of f; :: thesis: ( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) )
A3: (. (^ x)) . s = s . ((^ x) . s) by Def19;
((. x) mod (. (i,A,f))) . s = ((. x) . s) mod ((. (i,A,f)) . s) by Def30;
hence ( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds
(f . (s,(x %= i))) . z = s . z ) ) by A3, Th24; :: thesis: verum