
theorem Th16:
  for X,x,y being set holds X includes_lattice_of x, y iff x in X
  & y in X & x /\ y in X & x \/ y in X
proof
  let X,x,y be set;
  thus X includes_lattice_of x, y implies x in X & y in X & x /\ y in X & x \/
  y in X
  proof
A1: x \/ x = x & y \/ y = y;
A2: x in {x,y} & y in {x,y} by TARSKI:def 2;
    assume for a,b being set st a in {x,y} & b in {x,y} holds a /\ b in X & a
    \/ b in X;
    hence thesis by A2,A1;
  end;
  assume
A3: x in X & y in X & x /\ y in X & x \/ y in X;
  let a,b be set;
  assume that
A4: a in {x,y} and
A5: b in {x,y};
A6: b = x or b = y by A5,TARSKI:def 2;
  a = x or a = y by A4,TARSKI:def 2;
  hence thesis by A3,A6;
end;
