set f = I_RC ;
set g = I_LK ;
for x, y being Element of [.0,1.] holds I_RC . (x,y) <= I_LK . (x,y)

proof

let x, y be Element of [.0,1.]; :: thesis: I_RC . (x,y) <= I_LK . (x,y)

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

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

then I_LK . (x,y) = 1 by Lemma01;
hence I_RC . (x,y) <= I_LK . (x,y) by XXREAL_1:1; :: thesis: verum

end;

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

then C2: I_LK . (x,y) = (1 - x) + y by Lemma01;
C3: I_RC . (x,y) = (1 - x) + (x * y) by FUZIMPL1:def 17;
( y >= 0 & x <= 1 ) by XXREAL_1:1;
then x * y <= 1 * y by XREAL_1:64;
hence I_RC . (x,y) <= I_LK . (x,y) by C2, C3, XREAL_1:6; :: thesis: verum

end;

end;

end;

hence I_RC <= I_LK by FUZNORM1:def 16; :: thesis: verum