
theorem
  for L being with_suprema reflexive antisymmetric RelStr for D being
  Subset of L, x being Element of L st x is_<=_than D holds {x} "\/" D = D
proof
  let L be with_suprema reflexive antisymmetric RelStr, D be Subset of L, x be
  Element of L such that
A1: x is_<=_than D;
A2: {x} "\/" D = { x "\/" y where y is Element of L : y in D } by Th15;
  thus {x} "\/" D c= D
  proof
    let q be object;
    assume q in {x} "\/" D;
    then consider y being Element of L such that
A3: q = x "\/" y and
A4: y in D by A2;
    x <= y by A1,A4;
    hence thesis by A3,A4,YELLOW_0:24;
  end;
  let q be object;
  assume
A5: q in D;
  then reconsider D1 = D as non empty Subset of L;
  reconsider y = q as Element of D1 by A5;
  x <= y by A1;
  then q = x "\/" y by YELLOW_0:24;
  hence thesis by A2;
end;
