
theorem
  for L being with_suprema antisymmetric RelStr for D being Subset of L,
  x being Element of L holds x is_<=_than {x} "\/" D
proof
  let L be with_suprema antisymmetric RelStr, D be Subset of L, x be Element
  of L;
  let b be Element of L;
A1: {x} "\/" D = { x "\/" h where h is Element of L : h in D } by Th15;
  assume b in {x} "\/" D;
  then consider h being Element of L such that
A2: b = x "\/" h and
  h in D by A1;
  ex w being Element of L st x <= w & h <= w & for c being Element of L st
  x <= c & h <= c holds w <= c by LATTICE3:def 10;
  hence thesis by A2,LATTICE3:def 13;
end;
