reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;

theorem
  for X being non empty ext-real-membered set, x being UpperBound of X
  st x in X holds x = sup X
proof
  let X be non empty ext-real-membered set, x be UpperBound of X;
  assume x in X;
  then for y being UpperBound of X holds x <= y by Def1;
  hence thesis by Def3;
end;
