set f = I_FD ;

set g = I_LK ;

for x, y being Element of [.0,1.] holds I_FD . (x,y) <= I_LK . (x,y)

set g = I_LK ;

for x, y being Element of [.0,1.] holds I_FD . (x,y) <= I_LK . (x,y)

proof

hence
I_FD <= I_LK
by FUZNORM1:def 16; :: thesis: verum
let x, y be Element of [.0,1.]; :: thesis: I_FD . (x,y) <= I_LK . (x,y)

end;per cases
( x <= y or x > y )
;

end;

suppose B1:
x <= y
; :: thesis: I_FD . (x,y) <= I_LK . (x,y)

then
I_FD . (x,y) = 1
by FUZIMPL1:def 23;

hence I_FD . (x,y) <= I_LK . (x,y) by B1, Lemma01; :: thesis: verum

end;hence I_FD . (x,y) <= I_LK . (x,y) by B1, Lemma01; :: thesis: verum

suppose
x > y
; :: thesis: I_FD . (x,y) <= I_LK . (x,y)

then C3:
I_FD . (x,y) = max ((1 - x),y)
by FUZIMPL1:def 23;

C2: I_LK . (x,y) = min (1,((1 - x) + y)) by FUZIMPL1:def 14;

max ((1 - x),y) in [.0,1.] by Lemma03;

then C1: max ((1 - x),y) <= 1 by XXREAL_1:1;

( y >= 0 & x >= 0 ) by XXREAL_1:1;

then D1: (1 - x) + 0 <= (1 - x) + y by XREAL_1:6;

1 - x in [.0,1.] by FUZNORM1:7;

then 1 - x >= 0 by XXREAL_1:1;

then 0 + y <= (1 - x) + y by XREAL_1:6;

then max ((1 - x),y) <= (1 - x) + y by D1, XXREAL_0:28;

hence I_FD . (x,y) <= I_LK . (x,y) by C1, C2, C3, XXREAL_0:20; :: thesis: verum

end;C2: I_LK . (x,y) = min (1,((1 - x) + y)) by FUZIMPL1:def 14;

max ((1 - x),y) in [.0,1.] by Lemma03;

then C1: max ((1 - x),y) <= 1 by XXREAL_1:1;

( y >= 0 & x >= 0 ) by XXREAL_1:1;

then D1: (1 - x) + 0 <= (1 - x) + y by XREAL_1:6;

1 - x in [.0,1.] by FUZNORM1:7;

then 1 - x >= 0 by XXREAL_1:1;

then 0 + y <= (1 - x) + y by XREAL_1:6;

then max ((1 - x),y) <= (1 - x) + y by D1, XXREAL_0:28;

hence I_FD . (x,y) <= I_LK . (x,y) by C1, C2, C3, XXREAL_0:20; :: thesis: verum