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

proof

now :: thesis: (min ((f . c),(g . c))) * (min ((f . c),(h . c))) <= (min (f,(g * h))) . c

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

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

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

proof

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

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

A4: 0 <= g . c by Th1;
f . c <= h . c by A3, XXREAL_0:def 9;
then A5: (g . c) * (f . c) <= (h . c) * (g . c) by A4, XREAL_1:64;
A6: 0 <= f . c by Th1;
f c= by Th28;
then A7: (f * f) . c <= f . c ;
f . c <= g . c by A2, XXREAL_0:def 9;
then (f . c) * (f . c) <= (g . c) * (f . c) by A6, XREAL_1:64;
then (f . c) * (f . c) <= (g . c) * (h . c) by A5, XXREAL_0:2;
then min (((f * f) . c),((f . c) * (f . c))) <= min ((f . c),((g . c) * (h . c))) by A7, XXREAL_0:18;
then min (((f . c) * (f . c)),((f . c) * (f . c))) <= min ((f . c),((g . c) * (h . c))) by Def2;
then (f . c) * (f . c) <= min ((f . c),((g * h) . c)) by Def2;
hence (min ((f . c),(g . c))) * (min ((f . c),(h . c))) <= (min (f,(g * h))) . c by A2, A3, FUZZY_1:5; :: thesis: verum

end;

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

A9: 1 >= h . c by Th1;
A10: h . c >= 0 by Th1;
f . c <= g . c by A2, XXREAL_0:def 9;
then A11: (f . c) * (h . c) <= (g . c) * (h . c) by A10, XREAL_1:64;
f . c >= 0 by Th1;
then (f . c) * (h . c) <= (f . c) * 1 by A9, XREAL_1:64;
then (f . c) * (h . c) <= min ((f . c),((g . c) * (h . c))) by A11, XXREAL_0:20;
then (f . c) * (h . c) <= min ((f . c),((g * h) . c)) by Def2;
hence (min ((f . c),(g . c))) * (min ((f . c),(h . c))) <= (min (f,(g * h))) . c by A2, A8, FUZZY_1:5; :: thesis: verum

end;

end;

end;

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

end;

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

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

proof

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

suppose A13: min ((f . c),(h . c)) = f . c ; :: thesis: (min ((f . c),(g . c))) * (min ((f . c),(h . c))) <= (min (f,(g * h))) . c

A14: g . c >= 0 by Th1;
f . c <= h . c by A13, XXREAL_0:def 9;
then A15: (f . c) * (g . c) <= (h . c) * (g . c) by A14, XREAL_1:64;
A16: 1 >= g . c by Th1;
f . c >= 0 by Th1;
then (g . c) * (f . c) <= (f . c) * 1 by A16, XREAL_1:64;
then (f . c) * (g . c) <= min ((f . c),((h . c) * (g . c))) by A15, XXREAL_0:20;
then (f . c) * (g . c) <= min ((f . c),((g * h) . c)) by Def2;
hence (min ((f . c),(g . c))) * (min ((f . c),(h . c))) <= (min (f,(g * h))) . c by A12, A13, FUZZY_1:5; :: thesis: verum

end;

suppose A17: min ((f . c),(h . c)) = h . c ; :: thesis: (min ((f . c),(g . c))) * (min ((f . c),(h . c))) <= (min (f,(g * h))) . c

A18: g . c <= f . c by A12, XXREAL_0:def 9;
f . c >= 0 by Th1;
then A19: (g . c) * (f . c) <= (f . c) * (f . c) by A18, XREAL_1:64;
f c= by Th28;
then (f * f) . c <= f . c ;
then A20: (f . c) * (f . c) <= f . c by Def2;
A21: h . c <= f . c by A17, XXREAL_0:def 9;
g . c >= 0 by Th1;
then (h . c) * (g . c) <= (f . c) * (g . c) by A21, XREAL_1:64;
then (h . c) * (g . c) <= (f . c) * (f . c) by A19, XXREAL_0:2;
then (h . c) * (g . c) <= f . c by A20, XXREAL_0:2;
then (h . c) * (g . c) <= min ((f . c),((h . c) * (g . c))) by XXREAL_0:20;
then (g . c) * (h . c) <= min ((f . c),((g * h) . c)) by Def2;
hence (min ((f . c),(g . c))) * (min ((f . c),(h . c))) <= (min (f,(g * h))) . c by A12, A17, FUZZY_1:5; :: thesis: verum

end;

end;

end;

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

end;

end;

end;

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

end;

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