let A be Euclidean preIfWhileAlgebra; :: thesis: for X being non empty countable set
for s being Element of Funcs X,INT
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x being Variable of g holds
( (g . s,(x is_odd )) . b = (s . x) mod 2 & (g . s,(x is_even )) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(g . s,(x is_odd )) . z = s . z ) )
let X be non empty countable set ; :: thesis: for s being Element of Funcs X,INT
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x being Variable of g holds
( (g . s,(x is_odd )) . b = (s . x) mod 2 & (g . s,(x is_even )) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(g . s,(x is_odd )) . z = s . z ) )
let s be Element of Funcs X,INT ; :: thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x being Variable of g holds
( (g . s,(x is_odd )) . b = (s . x) mod 2 & (g . s,(x is_even )) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(g . s,(x is_odd )) . z = s . z ) )
let b be Element of X; :: thesis: for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x being Variable of g holds
( (g . s,(x is_odd )) . b = (s . x) mod 2 & (g . s,(x is_even )) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(g . s,(x is_odd )) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0 ; :: thesis: for x being Variable of f holds
( (f . s,(x is_odd )) . b = (s . x) mod 2 & (f . s,(x is_even )) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(f . s,(x is_odd )) . z = s . z ) )
let x be Variable of f; :: thesis: ( (f . s,(x is_odd )) . b = (s . x) mod 2 & (f . s,(x is_even )) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(f . s,(x is_odd )) . z = s . z ) )
( x is_odd = (. x) is_odd & (. x) . s = s . x ) by ThE1;
hence ( (f . s,(x is_odd )) . b = (s . x) mod 2 & (f . s,(x is_even )) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds
(f . s,(x is_odd )) . z = s . z ) ) by Th118; :: thesis: verum