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

theorem Th56:
  G is with_idempotent_element implies G is Huntington
proof
  assume G is with_idempotent_element;
  then consider C being Element of G such that
A1: C + C = C;
A2: now
    let x;
    thus C + x = -(-(-C + (C+x)) + -(C + (C+x))) by Def5
      .= -(-(-C + C+x) + -(C + (C+x))) by LATTICES:def 5
      .= -(-(C + -C + x) + -(C + x)) by A1,LATTICES:def 5;
  end;
  assume G is non Huntington;
  then consider B, A being Element of G such that
A3: -(-B + -A) + -(B + -A) <> A;
  set D = C + -C + -C;
A4: C = -(-C + -(C + -C)) by A1,Def5;
  then
A5: -(C + -(C + -C + -C)) = -C by Def5;
  then
  -(-C + -(C + -C + -C)) = -(-(-(C + -C + -C) + C) + -(C + C + (-C + -C)))
  by A1,LATTICES:def 5
    .= -(-(-(C + -C + -C) + C) + -(C + (C + (-C + -C)))) by LATTICES:def 5
    .= -(-(-D + C) + -(D + C)) by LATTICES:def 5
    .= C by Def5;
  then
A6: -(C + -C) = -(C + -C + -C) by A5,Def5;
  C = -(-(C + C) + -(-C + -(C + -C) + C)) by A4,Def5
    .= -(-C + -(C + -C + -(C + -C))) by A1,LATTICES:def 5;
  then
A7: C = C + -(C + -C) by A2,A5,A6;
A8: now
    let x;
    thus x = -(-(C + -(C + -C) + x) + -(-C + -(C + -C) + x)) by A4,A7,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;
A9: now
    let x;
    thus -(C + -C) = -(-(-x + -(C + -C)) + -(x + -(C + -C))) by Def5
      .= -(--x + -(x + -(C + -C))) by A8
      .= -(--x + -x) by A8;
  end;
A10: now
    let x;
    thus --x = -(-(-x + --x) + -(x + --x)) by Def5
      .= -(-(C + -C) + -(x + --x)) by A9
      .= --(x + --x) by A8;
  end;
A11: now
    let x;
    thus -x = -(-(---x + -x) + -(--x + -x)) by Def5
      .= -(-(---x + -x) + -(C + -C)) by A9
      .= --(---x + -x) by A8
      .= ---x by A10;
  end;
A12: now
    let x, y;
    thus y = -(-(-x + y) + -(x + y)) by Def5
      .= ---(-(-x + y) + -(x + y)) by A11
      .= --y by Def5;
  end;
  now
    let x, y;
    thus -(-x + y) + -(x + y) = --(-(-x + y) + -(x + y)) by A12
      .= -y by Def5;
  end;
  then -(-B + -A) + -(B + -A) = --A .= A by A12;
  hence thesis by A3;
end;
