 reserve W for WA-Lattice;
 reserve a,b,c for Element of W;

theorem :: Proposition 1
  for S being non empty RelStr holds
:::  st the InternalRel of S is antisymmetric holds
     S is tournament
      iff
    SymmetricHull S = nabla the carrier of S
  proof
    let S be non empty RelStr;
:::    assume the InternalRel of S is antisymmetric;
    thus S is tournament implies
    SymmetricHull S = nabla the carrier of S
    proof
      assume S is tournament; then
Aa1:  for a,b being Element of S holds
        a <= b or b <= a;
      for a,b being object holds
        [a,b] in SymmetricHull S
            implies
          [a,b] in nabla the carrier of S
      proof
        let a,b be object;
        assume [a,b] in SymmetricHull S; then
        [a,b] in [:the carrier of S, the carrier of S:];
        hence thesis by EQREL_1:def 1;
      end; then
A1:   SymmetricHull S c= nabla the carrier of S by RELAT_1:def 3;
      for a,b being object holds
        [a,b] in nabla the carrier of S implies
          [a,b] in SymmetricHull S
      proof
        let a,b be object;
        assume [a,b] in nabla the carrier of S; then
Z9:     [a,b] in [:the carrier of S,the carrier of S:]; then
        reconsider aa = a as Element of S by ZFMISC_1:87;
        reconsider bb = b as Element of S by Z9,ZFMISC_1:87;
        aa <= bb or bb <= aa by Aa1;
        hence [a,b] in SymmetricHull S by SymHull;
      end; then
      nabla the carrier of S c= SymmetricHull S by RELAT_1:def 3;
      hence thesis by A1,XBOOLE_0:def 10;
    end;
    assume
W0: SymmetricHull S = nabla the carrier of S;
    for x, y being Element of S holds
      x <= y or y <= x
    proof
      let x, y be Element of S;
      [x,y] in [:the carrier of S,the carrier of S:] by ZFMISC_1:87; then
      [x,y] in nabla the carrier of S by EQREL_1:def 1;
      hence thesis by W0,SymHull;
    end;
    hence thesis;
  end;
