reserve X for set;

theorem
  for X be non empty set st (for x,y be set st (x in X & y in X) holds x
  \/ y in X) holds InclPoset X is with_suprema
proof
  let X be non empty set;
  set L = InclPoset X;
  assume
A1: for x,y be set st x in X & y in X holds x \/ y in X;
  now
    let a,b be Element of L;
    ex c be Element of L st {a,b} is_<=_than c & for d be Element of L st
    {a,b} is_<=_than d holds c <= d
    proof
      take c = a "\/" b;
A2:   a \/ b = c by A1,Th8;
      then b c= c by XBOOLE_1:7;
      then
A3:   b <= c by Th3;
      a c= c by A2,XBOOLE_1:7;
      then a <= c by Th3;
      hence {a,b} is_<=_than c by A3,YELLOW_0:8;
      let d be Element of L;
      assume
A4:   {a,b} is_<=_than d;
      b in {a,b} by TARSKI:def 2;
      then b <= d by A4;
      then
A5:   b c= d by Th3;
      a in {a,b} by TARSKI:def 2;
      then a <= d by A4;
      then a c= d by Th3;
      then a \/ b c= d by A5,XBOOLE_1:8;
      then c c= d by A1,Th8;
      hence thesis by Th3;
    end;
    hence ex_sup_of {a,b}, L by YELLOW_0:15;
  end;
  hence thesis by YELLOW_0:20;
end;
