
theorem Th23:
  for L being naturally_sup-generated Lattice-like non empty
  LattRelStr holds the RelStr of L = LattPOSet L
proof
  let L be naturally_sup-generated Lattice-like non empty LattRelStr;
A1: for x, y being object holds [x,y] in the InternalRel of L iff [x,y] in
  LattRel L
  proof
    let x, y be object;
    hereby
      assume
A2:   [x,y] in the InternalRel of L;
      then reconsider x9 = x, y9 = y as Element of L by ZFMISC_1:87;
      x9 <= y9 by A2,ORDERS_2:def 5;
      then x9 [= y9 by Th22;
      hence [x,y] in LattRel L by FILTER_1:31;
    end;
    assume
A3: [x,y] in LattRel L;
    then reconsider x9 = x, y9 = y as Element of L by ZFMISC_1:87;
    x9 [= y9 by A3,FILTER_1:31;
    then x9 <= y9 by Th22;
    hence thesis by ORDERS_2:def 5;
  end;
  LattPOSet L = RelStr (#the carrier of L, LattRel L#) by LATTICE3:def 2;
  hence thesis by A1,RELAT_1:def 2;
end;
