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

theorem
  for f,g be RMembership_Func of C1,C2, h be RMembership_Func of C2,C3
  holds (max(f,g))(#)h = max(f(#)h,g(#)h)
proof
  let f,g be RMembership_Func of C1,C2, h be RMembership_Func of C2,C3;
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 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;
    ((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 thesis;
  end;
  dom((max(f,g))(#)h) = [:C1,C3:] by FUNCT_2:def 1;
  hence thesis by A1,A2,PARTFUN1:5;
end;
