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