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

theorem
  for F,G be Membership_Func of C1 holds (for x st x in C1 holds F.x <=
  G.x) implies upper_bound rng F <= upper_bound rng G
proof
  let F,G be Membership_Func of C1;
  rng F is real-bounded by Th1;
  then
A1: rng F is bounded_above by XXREAL_2:def 11;
  assume
A2: for c st c in C1 holds F.c <= G.c;
A3: for s being Real st 0<s ex y st y in dom F &
upper_bound rng F - s <= G.y
  proof
    let s being Real;
    assume 0<s;
    then consider r being Real such that
A4: r in rng F and
A5: upper_bound rng F-s<r by A1,SEQ_4:def 1;
    consider y being object such that
A6: y in dom F and
A7: r = F.y by A4,FUNCT_1:def 3;
    r <= G.y by A2,A6,A7;
    hence thesis by A5,A6,XXREAL_0:2;
  end;
  for s being Real st 0<s
  holds upper_bound rng F - s <= upper_bound rng G
  proof
    let s being Real;
    assume 0<s;
    then consider y such that
A8: y in dom F and
A9: upper_bound rng F - s <= G.y by A3;
    y in C1 by A8;
    then y in dom G by FUNCT_2:def 1;
    then G.y <= upper_bound rng G by Th1;
    hence thesis by A9,XXREAL_0:2;
  end;
  hence thesis by XREAL_1:57;
end;
