let F, G be Membership_Func of C; :: thesis: ( ( for c being Element of C holds F . c = (h . c) * (g . c) ) & ( for c being Element of C holds G . c = (h . c) * (g . c) ) implies F = G )
assume that
A11: for c being Element of C holds F . c = (h . c) * (g . c) and
A12: for c being Element of C holds G . c = (h . c) * (g . c) ; :: thesis: F = G
A13: ( C = dom F & C = dom G ) by FUNCT_2:def 1;
for c being Element of C st c in C holds
F . c = G . c

proof

let c be Element of C; :: thesis: ( c in C implies F . c = G . c )
( F . c = (h . c) * (g . c) & G . c = (h . c) * (g . c) ) by A11, A12;
hence ( c in C implies F . c = G . c ) ; :: thesis: verum

end;

hence F = G by A13, PARTFUN1:34; :: thesis: verum