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 number
for x being Variable of f holds
( (f . s,(x /= i)) . x = (s . x) div 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 number
for x being Variable of f holds
( (f . s,(x /= i)) . x = (s . x) div 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 number
for x being Variable of f holds
( (f . s,(x /= i)) . x = (s . x) div 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 number
for x being Variable of f holds
( (f . s,(x /= i)) . x = (s . x) div 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 number
for x being Variable of f holds
( (f . s,(x /= i)) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . s,(x /= i)) . z = s . z ) )
let i be integer number ; :: thesis: for x being Variable of f holds
( (f . s,(x /= i)) . x = (s . x) div 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) div i & ( for z being Element of X st z <> x holds
(f . s,(x /= i)) . z = s . z ) )
A1: (. i,A,f) . s = i by FUNCOP_1:13;
A2: (^ x) . s = x by FUNCOP_1:13;
A3: (. (^ x)) . s = s . ((^ x) . s) by Def19;
((. x) div (. i,A,f)) . s = ((. x) . s) div ((. i,A,f) . s) by Def29;
hence ( (f . s,(x /= i)) . x = (s . x) div i & ( for z being Element of X st z <> x holds
(f . s,(x /= i)) . z = s . z ) ) by A1, A2, A3, Th24; :: thesis: verum