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

proof

let c be Element of [:C1,C3:]; :: thesis: ( c in [:C1,C3:] implies (f (#) (max g,h)) . c = (max (f (#) g),(f (#) 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;
(f (#) (max g,h)) . c = (f (#) (max g,h)) . x,z by A5
.= sup (rng (min f,(max g,h),x,z)) by A5, Def3
.= max (sup (rng (min f,g,x,z))),(sup (rng (min f,h,x,z))) by A3, A4, Lm1
.= max ((f (#) g) . x,z),(sup (rng (min f,h,x,z))) by A5, Def3
.= max ((f (#) g) . x,z),((f (#) h) . x,z) by A5, Def3
.= (max (f (#) g),(f (#) h)) . c by A5, FUZZY_1:def 5 ;
hence ( c in [:C1,C3:] implies (f (#) (max g,h)) . c = (max (f (#) g),(f (#) h)) . c ) ; :: thesis: verum

end;

dom (f (#) (max g,h)) = [:C1,C3:] by FUNCT_2:def 1;
hence f (#) (max g,h) = max (f (#) g),(f (#) h) by A1, A2, PARTFUN1:34; :: thesis: verum