let C be non empty set ; :: thesis: for f, h, g being Membership_Func of C holds max (f,(g * h)) c=
let f, h, g be Membership_Func of C; :: thesis: max (f,(g * h)) c=
let c be Element of C; :: according to FUZZY_1:def 2 :: thesis: ((max (f,g)) * (max (f,h))) . c <= (max (f,(g * h))) . c
A1: (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c

proof

per cases ( max ((f . c),(g . c)) = f . c or max ((f . c),(g . c)) = g . c ) by XXREAL_0:16;

suppose A2: max ((f . c),(g . c)) = f . c ; :: thesis: (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c

(max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c

proof

per cases ( max ((f . c),(h . c)) = f . c or max ((f . c),(h . c)) = h . c ) by XXREAL_0:16;

suppose A3: max ((f . c),(h . c)) = f . c ; :: thesis: (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c

(max (f,(g * h))) . c = max ((f . c),((g * h) . c)) by FUZZY_1:5;
then A4: (max (f,(g * h))) . c >= f . c by XXREAL_0:25;
f c= by Th28;
then (f * f) . c <= f . c ;
then (f * f) . c <= (max (f,(g * h))) . c by A4, XXREAL_0:2;
hence (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c by A2, A3, Def2; :: thesis: verum

end;

suppose A5: max ((f . c),(h . c)) = h . c ; :: thesis: (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c

A6: 1 >= h . c by Th1;
f . c <= max ((f . c),((g * h) . c)) by XXREAL_0:25;
then A7: f . c <= (max (f,(g * h))) . c by FUZZY_1:5;
f . c >= 0 by Th1;
then (f . c) * (h . c) <= (f . c) * 1 by A6, XREAL_1:64;
hence (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c by A2, A5, A7, XXREAL_0:2; :: thesis: verum

end;

end;

end;

hence (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c ; :: thesis: verum

end;

suppose A8: max ((f . c),(g . c)) = g . c ; :: thesis: (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c

(max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c

proof

per cases ( max ((f . c),(h . c)) = f . c or max ((f . c),(h . c)) = h . c ) by XXREAL_0:16;

suppose A9: max ((f . c),(h . c)) = f . c ; :: thesis: (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c

A10: 1 >= g . c by Th1;
f . c <= max ((f . c),((g * h) . c)) by XXREAL_0:25;
then A11: f . c <= (max (f,(g * h))) . c by FUZZY_1:5;
f . c >= 0 by Th1;
then (f . c) * (g . c) <= (f . c) * 1 by A10, XREAL_1:64;
hence (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c by A8, A9, A11, XXREAL_0:2; :: thesis: verum

end;

suppose A12: max ((f . c),(h . c)) = h . c ; :: thesis: (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c

(max (f,(g * h))) . c = max ((f . c),((g * h) . c)) by FUZZY_1:5;
then (max (f,(g * h))) . c >= (g * h) . c by XXREAL_0:25;
hence (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c by A8, A12, Def2; :: thesis: verum

end;

end;

end;

hence (max ((f . c),(g . c))) * (max ((f . c),(h . c))) <= (max (f,(g * h))) . c ; :: thesis: verum

end;

end;

end;

((max (f,g)) * (max (f,h))) . c = ((max (f,g)) . c) * ((max (f,h)) . c) by Def2
.= (max ((f . c),(g . c))) * ((max (f,h)) . c) by FUZZY_1:5
.= (max ((f . c),(g . c))) * (max ((f . c),(h . c))) by FUZZY_1:5 ;
hence ((max (f,g)) * (max (f,h))) . c <= (max (f,(g * h))) . c by A1; :: thesis: verum