let L be complete Lattice; :: thesis: for f being monotone UnOp of L
for x, y being Element of
for a, b being Element of st x = a & y = b holds
( x [= y iff a [= b )
let f be monotone UnOp of L; :: thesis: for x, y being Element of
for a, b being Element of st x = a & y = b holds
( x [= y iff a [= b )
A1: ex P being non empty with_suprema with_infima Subset of st
( P = { x where x is Element of : x is_a_fixpoint_of f } & FixPoints f = latt P ) by Def11;
let x, y be Element of ; :: thesis: for a, b being Element of st x = a & y = b holds
( x [= y iff a [= b )
let a, b be Element of ; :: thesis: ( x = a & y = b implies ( x [= y iff a [= b ) )
assume A2: ( x = a & y = b ) ; :: thesis: ( x [= y iff a [= b )
ex a', b' being Element of st
( x = a' & y = b' & ( x [= y implies a' [= b' ) & ( a' [= b' implies x [= y ) ) by A1, Def10;
hence ( x [= y iff a [= b ) by A2; :: thesis: verum