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(#)(min(g,h)) c= min(f(#)g,f(#)h)
proof
  let f be RMembership_Func of C1,C2,g,h be RMembership_Func of C2,C3;
  let c be Element of [:C1,C3:];
  consider x,z being object such that
A1: x in C1 and
A2: z in C3 and
A3: c = [x,z] by ZFMISC_1:def 2;
    reconsider z,x as set by TARSKI:1;
A4: (f(#)(min(g,h))).(x,z) = upper_bound(rng(min(f,min(g,h),x,z))) by A3,Def3;
  min(f(#)g,f(#)h).(x,z) = min((f(#)g).(x,z),(f(#)h).(x,z)) by A3,FUZZY_1:def 3
    .= min((f(#)g).(x,z),upper_bound rng(min(f,h,x,z))) by A3,Def3
    .= min(upper_bound rng(min(f,g,x,z)),
    upper_bound rng(min(f,h,x,z))) by A3,Def3;
  hence thesis by A1,A2,A3,A4,Lm3;
end;
