
theorem Th31:
  for k being Element of NAT, a, b being Element of SubstPoset (
  NAT, {k}), X being Subset of SubstPoset (NAT, {k}) st a is_<=_than X & b
  is_<=_than X holds a "\/" b is_<=_than X
proof
  let k be Element of NAT;
  let a, b be Element of SubstPoset (NAT, {k}), X be Subset of SubstPoset (NAT
  , {k});
  assume
A1: a is_<=_than X & b is_<=_than X;
  let c be Element of SubstPoset (NAT, {k});
  assume c in X;
  then a <= c & b <= c by A1;
  hence thesis by YELLOW_5:9;
end;
