
theorem
  for L being non empty RelStr, D being Subset of [:L,L:] holds union {X
where X is Subset of L: ex x being Element of L st X = {x} "\/" proj2 D & x in
  proj1 D} = proj1 D "\/" proj2 D
proof
  let L be non empty RelStr, D be Subset of [:L,L:];
  set D1 = proj1 D, D2 = proj2 D;
  defpred P[set] means ex x being Element of L st $1 = {x} "\/" proj2 D & x in
  proj1 D;
  thus union {X where X is Subset of L: P[X]} c= D1 "\/" D2
  proof
    let q be object;
    assume q in union {X where X is Subset of L: P[X]};
    then consider w being set such that
A1: q in w and
A2: w in {X where X is Subset of L: P[X]} by TARSKI:def 4;
    consider e being Subset of L such that
A3: w = e and
A4: P[e] by A2;
    consider p being Element of L such that
A5: e = {p} "\/" D2 and
A6: p in D1 by A4;
    {p} "\/" D2 = { p "\/" y where y is Element of L : y in D2 } by Th15;
    then ex y being Element of L st q = p "\/" y & y in D2 by A1,A3,A5;
    hence thesis by A6;
  end;
  let q be object;
  assume q in D1 "\/" D2;
  then consider x, y being Element of L such that
A7: q = x "\/" y and
A8: x in D1 and
A9: y in D2;
  {x} "\/" D2 = { x "\/" d where d is Element of L : d in D2 } by Th15;
  then
A10: q in {x} "\/" D2 by A7,A9;
  {x} "\/" D2 in {X where X is Subset of L: P[X]} by A8;
  hence thesis by A10,TARSKI:def 4;
end;
