
theorem
  for X be non empty set for Y be non empty Subset of InclPoset X st
  ex_sup_of Y,InclPoset X holds union Y c= sup Y
proof
  let X be non empty set;
  let Y be non empty Subset of InclPoset X;
  assume
A1: ex_sup_of Y,InclPoset X;
  now
    let x be set;
    assume
A2: x in Y;
    then reconsider x1 = x as Element of InclPoset X;
    sup Y is_>=_than Y by A1,YELLOW_0:30;
    then sup Y >= x1 by A2,LATTICE3:def 9;
    hence x c= sup Y by YELLOW_1:3;
  end;
  hence thesis by ZFMISC_1:76;
end;
