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 gt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x gt y) in (Funcs X,INT ) \ b,0 implies s . x > s . y ) & ( s . x < s . y implies g . s,(x lt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x lt y) in (Funcs X,INT ) \ b,0 implies s . x < s . y ) )
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 gt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x gt y) in (Funcs X,INT ) \ b,0 implies s . x > s . y ) & ( s . x < s . y implies g . s,(x lt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x lt y) in (Funcs X,INT ) \ b,0 implies s . x < s . y ) )
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 gt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x gt y) in (Funcs X,INT ) \ b,0 implies s . x > s . y ) & ( s . x < s . y implies g . s,(x lt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x lt y) in (Funcs X,INT ) \ b,0 implies s . x < s . y ) )
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 gt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x gt y) in (Funcs X,INT ) \ b,0 implies s . x > s . y ) & ( s . x < s . y implies g . s,(x lt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x lt y) in (Funcs X,INT ) \ b,0 implies s . x < s . y ) )
let g be Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0 ; :: thesis: for x, y being Variable of g holds
( ( s . x > s . y implies g . s,(x gt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x gt y) in (Funcs X,INT ) \ b,0 implies s . x > s . y ) & ( s . x < s . y implies g . s,(x lt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x lt y) in (Funcs X,INT ) \ b,0 implies s . x < s . y ) )
let x, y be Variable of g; :: thesis: ( ( s . x > s . y implies g . s,(x gt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x gt y) in (Funcs X,INT ) \ b,0 implies s . x > s . y ) & ( s . x < s . y implies g . s,(x lt y) in (Funcs X,INT ) \ b,0 ) & ( g . s,(x lt y) in (Funcs X,INT ) \ b,0 implies s . x < s . y ) )
( g . s,(x gt y) in (Funcs X,INT ) \ b,0 iff (g . s,(x gt y)) . b <> 0 ) by Th2;
hence ( s . x > s . y iff g . s,(x gt y) in (Funcs X,INT ) \ b,0 ) by Th39; :: thesis: ( s . x < s . y iff g . s,(x lt y) in (Funcs X,INT ) \ b,0 )
( g . s,(x lt y) in (Funcs X,INT ) \ b,0 iff (g . s,(x lt y)) . b <> 0 ) by Th2;
hence ( s . x < s . y iff g . s,(x lt y) in (Funcs X,INT ) \ b,0 ) by Th39; :: thesis: verum