let S, T be non empty RelStr ; :: thesis: for f being Function of S,(T opp)
for g being Function of (S opp),T st f = g holds
( ( f is monotone implies g is monotone ) & ( g is monotone implies f is monotone ) & ( f is antitone implies g is antitone ) & ( g is antitone implies f is antitone ) )
let f be Function of S,(T ~); :: thesis: for g being Function of (S opp),T st f = g holds
( ( f is monotone implies g is monotone ) & ( g is monotone implies f is monotone ) & ( f is antitone implies g is antitone ) & ( g is antitone implies f is antitone ) )
let g be Function of (S ~),T; :: thesis: ( f = g implies ( ( f is monotone implies g is monotone ) & ( g is monotone implies f is monotone ) & ( f is antitone implies g is antitone ) & ( g is antitone implies f is antitone ) ) )
assume A1: f = g ; :: thesis: ( ( f is monotone implies g is monotone ) & ( g is monotone implies f is monotone ) & ( f is antitone implies g is antitone ) & ( g is antitone implies f is antitone ) )
thus ( f is monotone implies g is monotone ) :: thesis: ( ( g is monotone implies f is monotone ) & ( f is antitone implies g is antitone ) & ( g is antitone implies f is antitone ) )

proof

assume A2: for x, y being Element of S st x <= y holds
f . x <= f . y ; :: according to WAYBEL_1:def 2 :: thesis: g is monotone
let x, y be Element of (S ~); :: according to WAYBEL_1:def 2 :: thesis: ( not x <= y or g . x <= g . y )
assume x <= y ; :: thesis: g . x <= g . y
then ~ y <= ~ x by Th1;
then A3: f . (~ y) <= f . (~ x) by A2;
( ~ (f . (~ x)) = f . (~ x) & ~ (f . (~ y)) = f . (~ y) ) ;
hence g . x <= g . y by A1, A3, Th1; :: thesis: verum

end;

thus ( g is monotone implies f is monotone ) :: thesis: ( f is antitone iff g is antitone )

proof

assume A4: for x, y being Element of (S opp) st x <= y holds
g . x <= g . y ; :: according to WAYBEL_1:def 2 :: thesis: f is monotone
let x, y be Element of S; :: according to WAYBEL_1:def 2 :: thesis: ( not x <= y or f . x <= f . y )
assume x <= y ; :: thesis: f . x <= f . y
then y ~ <= x ~ by LATTICE3:9;
then A5: g . (y ~) <= g . (x ~) by A4;
( (g . (x ~)) ~ = g . (x ~) & (g . (y ~)) ~ = g . (y ~) ) ;
hence f . x <= f . y by A1, A5, LATTICE3:9; :: thesis: verum

end;

thus ( f is antitone implies g is antitone ) :: thesis: ( g is antitone implies f is antitone )

proof

assume A6: for x, y being Element of S st x <= y holds
for a, b being Element of (T ~) st a = f . x & b = f . y holds
a >= b ; :: according to WAYBEL_0:def 5 :: thesis: g is antitone
let x, y be Element of (S ~); :: according to WAYBEL_0:def 5 :: thesis: ( not x <= y or for b₁, b₂ being Element of the carrier of T holds
( not b₁ = g . x or not b₂ = g . y or b₂ <= b₁ ) )
assume x <= y ; :: thesis: for b₁, b₂ being Element of the carrier of T holds
( not b₁ = g . x or not b₂ = g . y or b₂ <= b₁ )
then ~ y <= ~ x by Th1;
then A7: f . (~ y) >= f . (~ x) by A6;
( ~ (f . (~ x)) = f . (~ x) & ~ (f . (~ y)) = f . (~ y) ) ;
hence for b₁, b₂ being Element of the carrier of T holds
( not b₁ = g . x or not b₂ = g . y or b₂ <= b₁ ) by A1, A7, Th1; :: thesis: verum

end;

thus ( g is antitone implies f is antitone ) :: thesis: verum

proof

assume A8: for x, y being Element of (S opp) st x <= y holds
for a, b being Element of T st a = g . x & b = g . y holds
a >= b ; :: according to WAYBEL_0:def 5 :: thesis: f is antitone
let x, y be Element of S; :: according to WAYBEL_0:def 5 :: thesis: ( not x <= y or for b₁, b₂ being Element of the carrier of (T ~) holds
( not b₁ = f . x or not b₂ = f . y or b₂ <= b₁ ) )
assume x <= y ; :: thesis: for b₁, b₂ being Element of the carrier of (T ~) holds
( not b₁ = f . x or not b₂ = f . y or b₂ <= b₁ )
then y ~ <= x ~ by LATTICE3:9;
then A9: g . (y ~) >= g . (x ~) by A8;
( (g . (x ~)) ~ = g . (x ~) & (g . (y ~)) ~ = g . (y ~) ) ;
hence for b₁, b₂ being Element of the carrier of (T ~) holds
( not b₁ = f . x or not b₂ = f . y or b₂ <= b₁ ) by A1, A9, LATTICE3:9; :: thesis: verum

end;