let C1, C2, C3 be non empty set ; :: thesis: for f, g being RMembership_Func of C1,C2
for h being RMembership_Func of C2,C3 holds (max (f,g)) (#) h = max ((f (#) h),(g (#) h))
let f, g be RMembership_Func of C1,C2; :: thesis: for h being RMembership_Func of C2,C3 holds (max (f,g)) (#) h = max ((f (#) h),(g (#) h))
let h be RMembership_Func of C2,C3; :: thesis: (max (f,g)) (#) h = max ((f (#) h),(g (#) h))
A1: dom (max ((f (#) h),(g (#) h))) = [:C1,C3:] by FUNCT_2:def 1;
A2: for c being Element of [:C1,C3:] st c in [:C1,C3:] holds
((max (f,g)) (#) h) . c = (max ((f (#) h),(g (#) h))) . c

proof

let c be Element of [:C1,C3:]; :: thesis: ( c in [:C1,C3:] implies ((max (f,g)) (#) h) . c = (max ((f (#) h),(g (#) h))) . c )
consider x, z being object such that
A3: x in C1 and
A4: z in C3 and
A5: c = [x,z] by ZFMISC_1:def 2;
reconsider z = z, x = x as set by TARSKI:1;
((max (f,g)) (#) h) . c = ((max (f,g)) (#) h) . (x,z) by A5
.= upper_bound (rng (min ((max (f,g)),h,x,z))) by A5, Def3
.= max ((upper_bound (rng (min (f,h,x,z)))),(upper_bound (rng (min (g,h,x,z))))) by A3, A4, Lm2
.= max (((f (#) h) . (x,z)),(upper_bound (rng (min (g,h,x,z))))) by A5, Def3
.= max (((f (#) h) . (x,z)),((g (#) h) . (x,z))) by A5, Def3
.= (max ((f (#) h),(g (#) h))) . c by A5, FUZZY_1:def 4 ;
hence ( c in [:C1,C3:] implies ((max (f,g)) (#) h) . c = (max ((f (#) h),(g (#) h))) . c ) ; :: thesis: verum

end;

dom ((max (f,g)) (#) h) = [:C1,C3:] by FUNCT_2:def 1;
hence (max (f,g)) (#) h = max ((f (#) h),(g (#) h)) by A1, A2, PARTFUN1:5; :: thesis: verum