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, y being Variable of g holds
( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(g . (s,(x leq y))) . 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, y being Variable of g holds
( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(g . (s,(x leq y))) . 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, y being Variable of g holds
( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(g . (s,(x leq y))) . 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, y being Variable of g holds
( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(g . (s,(x leq y))) . z = s . z ) )
let f be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); :: thesis: for x, y being Variable of f holds
( ( s . x <= s . y implies (f . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (f . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(f . (s,(x leq y))) . z = s . z ) )
reconsider b9 = b as Variable of f by Def2;
let x, y be Variable of f; :: thesis: ( ( s . x <= s . y implies (f . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (f . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(f . (s,(x leq y))) . z = s . z ) )
set v = ^ b9;
A1: (^ b9) . s = b ;
A2: ( (. x) . s <= (. y) . s implies IFGT (((. x) . s),((. y) . s),0,1) = 1 ) by XXREAL_0:def 11;
(. x) . s = s . ((^ x) . s) by Def19;
then A3: s . x = (. x) . s ;
A4: ( (. x) . s > (. y) . s implies IFGT (((. x) . s),((. y) . s),0,1) = 0 ) by XXREAL_0:def 11;
A5: (leq ((. x),(. y))) . s = IFGT (((. x) . s),((. y) . s),0,1) by Def31;
(. y) . s = s . ((^ y) . s) by Def19;
hence ( ( s . x <= s . y implies (f . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (f . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds
(f . (s,(x leq y))) . z = s . z ) ) by A3, A1, A2, A4, A5, Th24; :: thesis: verum