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 (f (#) (max g,h)) = [:C1,C3:] & dom (f (#) h) = [:C1,C3:] & dom (max (f (#) g),(f (#) h)) = [:C1,C3:] & dom (f (#) g) = [:C1,C3:] ) by FUNCT_2:def 1;
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
A2: ( x in C1 & z in C3 & c = [x,z] ) by ZFMISC_1:def 2;
(f (#) (max g,h)) . c = (f (#) (max g,h)) . x,z by A2
.= sup (rng (min f,(max g,h),x,z)) by A2, Def3
.= max (sup (rng (min f,g,x,z))),(sup (rng (min f,h,x,z))) by A2, Lm1
.= max ((f (#) g) . x,z),(sup (rng (min f,h,x,z))) by A2, Def3
.= max ((f (#) g) . x,z),((f (#) h) . x,z) by A2, Def3
.= (max (f (#) g),(f (#) h)) . c by A2, FUZZY_1:def 5 ;
hence ( c in [:C1,C3:] implies (f (#) (max g,h)) . c = (max (f (#) g),(f (#) h)) . c ) ; :: thesis: verum

end;

hence f (#) (max g,h) = max (f (#) g),(f (#) h) by A1, PARTFUN1:34; :: thesis: verum