let C be non empty set ; :: thesis: for f, g being Membership_Func of C holds f ++ g c=
let f, g be Membership_Func of C; :: thesis: f ++ g c=
let c be Element of C; :: according to FUZZY_1:def 3 :: thesis: (max f,g) . c <= (f ++ g) . c
A1: (max f,g) . c = max (f . c),(g . c) by FUZZY_1:6;

per cases ( (max f,g) . c = f . c or (max f,g) . c = g . c ) by A1, XXREAL_0:16;

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

A3: (1_minus f) . c >= 0 by Th1;
g . c >= 0 by Th1;
then 0 * (g . c) <= (g . c) * ((1_minus f) . c) by A3, XREAL_1:66;
then 0 <= (g . c) * (1 - (f . c)) by FUZZY_1:def 6;
then 0 + (f . c) <= ((g . c) - ((f . c) * (g . c))) + (f . c) by XREAL_1:8;
then f . c <= ((f . c) + (g . c)) - ((f . c) * (g . c)) ;
hence (max f,g) . c <= (f ++ g) . c by A2, Def3; :: thesis: verum

end;

suppose A4: (max f,g) . c = g . c ; :: thesis: (max f,g) . c <= (f ++ g) . c

A5: (1_minus g) . c >= 0 by Th1;
f . c >= 0 by Th1;
then 0 * (f . c) <= (f . c) * ((1_minus g) . c) by A5, XREAL_1:66;
then 0 <= (f . c) * (1 - (g . c)) by FUZZY_1:def 6;
then 0 + (g . c) <= ((f . c) - ((f . c) * (g . c))) + (g . c) by XREAL_1:8;
then g . c <= ((f . c) + (g . c)) - ((f . c) * (g . c)) ;
hence (max f,g) . c <= (f ++ g) . c by A4, Def3; :: thesis: verum

end;

end;