reserve x,y,z,r,s for ExtReal;

theorem Th1:
  y is UpperBound of {x} iff x <= y
proof
  x in {x} by TARSKI:def 1;
  hence y is UpperBound of {x} implies x <= y by Def1;
  assume
A1: x <= y;
  let z;
  assume z in {x};
  hence thesis by A1,TARSKI:def 1;
end;
