let C be non empty set ; :: thesis: for f, g, h being Membership_Func of C holds (min f,g) ++ (min f,h) c=
let f, g, h be Membership_Func of C; :: thesis: (min f,g) ++ (min f,h) c=
let c be Element of C; :: according to FUZZY_1:def 3 :: thesis: (min f,(g ++ h)) . c <= ((min f,g) ++ (min f,h)) . c
A1: ((min f,g) ++ (min f,h)) . c = (((min f,g) . c) + ((min f,h) . c)) - (((min f,g) . c) * ((min f,h) . c)) by Def3
.= ((min (f . c),(g . c)) + ((min f,h) . c)) - (((min f,g) . c) * ((min f,h) . c)) by FUZZY_1:6
.= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - (((min f,g) . c) * ((min f,h) . c)) by FUZZY_1:6
.= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * ((min f,h) . c)) by FUZZY_1:6
.= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c))) by FUZZY_1:6 ;
A2: min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c)))

proof

now

per cases ( ( min (f . c),(g . c) = f . c & min (f . c),(h . c) = f . c ) or ( min (f . c),(g . c) = f . c & min (f . c),(h . c) = h . c ) or ( min (f . c),(g . c) = g . c & min (f . c),(h . c) = f . c ) or ( min (f . c),(g . c) = g . c & min (f . c),(h . c) = h . c ) ) by XXREAL_0:15;

suppose A3: ( min (f . c),(g . c) = f . c & min (f . c),(h . c) = f . c ) ; :: thesis: min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c)))

f ++ f c= by Th33;
then A4: (f ++ f) . c >= f . c by FUZZY_1:def 3;
min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= f . c by XXREAL_0:17;
then min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= (f ++ f) . c by A4, XXREAL_0:2;
hence min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c))) by A3, Def3; :: thesis: verum

end;

suppose A5: ( min (f . c),(g . c) = f . c & min (f . c),(h . c) = h . c ) ; :: thesis: min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c)))

(1_minus f) . c >= 0 by Th1;
then A6: 1 - (f . c) >= 0 by FUZZY_1:def 6;
h . c >= 0 by Th1;
then 0 * (h . c) <= (h . c) * (1 - (f . c)) by A6, XREAL_1:66;
then A7: 0 + (f . c) <= ((h . c) * (1 - (f . c))) + (f . c) by XREAL_1:8;
min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= f . c by XXREAL_0:17;
hence min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c))) by A5, A7, XXREAL_0:2; :: thesis: verum

end;

suppose A8: ( min (f . c),(g . c) = g . c & min (f . c),(h . c) = f . c ) ; :: thesis: min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c)))

(1_minus f) . c >= 0 by Th1;
then A9: 1 - (f . c) >= 0 by FUZZY_1:def 6;
g . c >= 0 by Th1;
then 0 * (g . c) <= (g . c) * (1 - (f . c)) by A9, XREAL_1:66;
then A10: 0 + (f . c) <= ((g . c) * (1 - (f . c))) + (f . c) by XREAL_1:8;
min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= f . c by XXREAL_0:17;
hence min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c))) by A8, A10, XXREAL_0:2; :: thesis: verum

end;

suppose ( min (f . c),(g . c) = g . c & min (f . c),(h . c) = h . c ) ; :: thesis: min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c)))

hence min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c))) by XXREAL_0:17; :: thesis: verum

end;

end;

end;

hence min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) <= ((min (f . c),(g . c)) + (min (f . c),(h . c))) - ((min (f . c),(g . c)) * (min (f . c),(h . c))) ; :: thesis: verum

end;

(min f,(g ++ h)) . c = min (f . c),((g ++ h) . c) by FUZZY_1:6
.= min (f . c),(1 - ((1 - (g . c)) * (1 - (h . c)))) by Th54 ;
hence (min f,(g ++ h)) . c <= ((min f,g) ++ (min f,h)) . c by A1, A2; :: thesis: verum