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) 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

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

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

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

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; :: 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 ) )

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 A3, Th24; :: thesis: verum

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) 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

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

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

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

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; :: 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 ) )

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 A3, Th24; :: thesis: verum