reserve G for Robbins join-associative join-commutative non empty
  ComplLLattStr;
reserve x, y, z, u, v for Element of G;

theorem Th57:
  (ex c, d being Element of G st c + d = c) implies G is Huntington
proof
A1: now
    let x, y, z;
    set k = -(-x + y);
    thus -(-(-(-x + y) + x + y) + y) = -(-(k + x + y) + -(k + -(x + y))) by
Def5
      .= -(-(k + (x + y)) + -(k + -(x + y))) by LATTICES:def 5
      .= -(-x + y) by Def5;
  end;
A2: now
    let x, y, z;
    set k = -(-x + y);
    -(-(k + x + y) + y) = k by A1;
    hence z = -(-(-(-(-x + y) + x + y) + y + z) + -(-(-x + y) + z)) by Def5;
  end;
  given C, D being Element of G such that
A3: C + D = C;
A4: now
    let x, y, z;
    set k = -(-x + y) + -(x + y);
    thus -(-(-(-x + y) + -(x + y) + z) + -(y + z)) = -(-(k + z) + -(-k + z))
    by Def5
      .= z by Def5;
  end;
A5: now
    let x;
    thus D = -(-(-(-x + C) + -(x + C) + D) + -(C + D)) by A4
      .= -(-C + -(D + -(C + -x) + -(C + x))) by A3,LATTICES:def 5;
  end;
  set e = -(C + -C);
  set K = -(C + C + -(C + -C));
A6: K = -(C + -(C + -C) + C) by LATTICES:def 5;
A7: now
    let x, y;
    thus -(-(D + -(C + x) + y) + -(C + x + y)) = -(-(-(C + x) + (D + y)) + -(C
    + D + x + y)) by A3,LATTICES:def 5
      .= -(-(-(C + x) + (D + y)) + -(C + x + D + y)) by LATTICES:def 5
      .= -(-(-(C + x) + (D + y)) + -((C + x) + (D + y))) by LATTICES:def 5
      .= D + y by Def5;
  end;
  set L = -(D + -C);
  set E = D + -C;
A8: -(-C + -(D + -C)) = D by A3,Def5;
  then
A9: -(D + -(C + -E)) = -E by Def5;
  -(L + -(C + L)) = -(-(D + -(C + L)) + -((D + C) + L)) by A3,A8,Def5
    .= -(-(D + -(C + L)) + -(D + (C + L))) by LATTICES:def 5
    .= D by Def5;
  then -(D + -(D + -C + -(C + -(D + -C)))) = -(C + -(D + -C)) by Def5;
  then
A10: -(C + -(D + -C)) = -(D + -(D + -(C + -(D + -C)) + -C)) by LATTICES:def 5
    .= -C by A8,A9,Def5;
  set L = C + -(D + -C);
A11: C = -(-(D + -L + C) + -(-(D + -C) + C)) by A9,Def5
    .= -(-L + -(C + -L)) by A3,LATTICES:def 5;
  then -(C + -(C + -(C + -C))) = -(C + -C) by A10,Def5;
  then C = -(-(C + -C) + K) by A6,Def5;
  then -(C + -(C + -C + K)) = K by Def5;
  then K = -(-(K + C + -C) + C) by LATTICES:def 5
    .= -(-(-(-(C + -C) + C + C) + C + -C) + C) by LATTICES:def 5
    .= -C by A11,A2,A10;
  then D + -(C + -C) = -(-(D + -(C + C) + -(C + -C)) + -C) by A7
    .= -(-C + -(D + -(C + -C) + -(C + C))) by LATTICES:def 5
    .= D by A5;
  then
A12: C + -(C + -C) = C by A3,LATTICES:def 5;
  now
    let x;
    thus x = -(-(C + -(C + -C) + x) + -(-C + -(C + -C) + x)) by A11,A10,A12
,Def5
      .= -(-(C + (-(C + -C) + x)) + -(-C + -(C + -C) + x)) by LATTICES:def 5
      .= -(-(C + (-(C + -C) + x)) + -(-C + (-(C + -C) + x))) by LATTICES:def 5
      .= -(C + -C) + x by Def5;
  end;
  then e = e + e;
  then G is with_idempotent_element;
  hence thesis;
end;
