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

proof

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

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

suppose C1: x <= y ; :: thesis: I_GG . (x,y) <= I_LK . (x,y)

then I_LK . (x,y) = 1 by Lemma01;
hence I_GG . (x,y) <= I_LK . (x,y) by C1, FUZIMPL1:def 19; :: thesis: verum

end;

suppose C1: x > y ; :: thesis: I_GG . (x,y) <= I_LK . (x,y)

then C2: I_GG . (x,y) = y / x by FUZIMPL1:def 19;
I_LK . (x,y) = (1 - x) + y by C1, Lemma01;
hence I_GG . (x,y) <= I_LK . (x,y) by C2, C1, Lemacik1; :: thesis: verum

end;

end;

end;

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