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 set such that
A3: x in C1 and
A4: z in C3 and
A5: c = [x,z] by ZFMISC_1:def 2;
((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 5 ;
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, PARTFUN1:34; :: thesis: verum