reserve c,c1,c2,x,y,z,z1,z2 for set;
reserve C1,C2,C3 for non empty set;

theorem
  for f be RMembership_Func of C1,C2, g,h be RMembership_Func of C2,C3
  holds f(#)(max(g,h)) = max(f(#)g,f(#)h)
proof
  let f be RMembership_Func of C1,C2,g,h be RMembership_Func of C2,C3;
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 being Element of [:C1,C3:];
    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,x as set by TARSKI:1;
    (f(#)(max(g,h))).c = (f(#)(max(g,h))).(x,z) by A5
      .= upper_bound(rng(min(f,max(g,h),x,z))) by A5,Def3
      .= max(upper_bound rng(min(f,g,x,z)),upper_bound rng(min(f,h,x,z)))
      by A3,A4,Lm1
      .= max((f(#)g).(x,z),upper_bound 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 4;
    hence thesis;
  end;
  dom(f(#)(max(g,h))) = [:C1,C3:] by FUNCT_2:def 1;
  hence thesis by A1,A2,PARTFUN1:5;
end;
