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

theorem Th58:
  ex y, z st y + z = z
proof
A1: now
    let x, y;
    thus -(x + y) = -(-(-(-x + y) + -(x + y)) + -(-x + y + -(x + y))) by Def5
      .= -(y + -(-x + y + -(x + y))) by Def5
      .= -(-(-(x + y) + -x + y) + y) by LATTICES:def 5;
  end;
A2: now
    let x, y;
    thus -(-x + y) = -(-(-(x + y) + -(-x + y)) + -((x + y) + -(-x + y))) by
Def5
      .= -(y + -(x + y + -(-x + y))) by Def5
      .= -(-(-(-x + y) + x + y) + y) by LATTICES:def 5;
  end;
A3: now
    let x, y;
    thus y = -(-(-(-(-x + y) + x + y) + y) + -((-(-x + y) + x + y) + y)) by
Def5
      .= -(-(-x + y) + -((-(-x + y) + x + y) + y)) by A2
      .= -(-(-(-x + y) + x + Double y) + -(-x + y)) by LATTICES:def 5;
  end;
A4: now
    let x, y, z;
    thus z = -(-(-(-(-(-x + y) + x + Double y) + -(-x + y)) + z) + -(-(-(-x +
    y) + x + Double y) + -(-x + y) + z)) by Def5
      .= -(-(-(-(-x + y) + x + Double y) + -(-x + y) + z) + -(y + z)) by A3;
  end;
A5: now
    let x, y, z;
    set k = -(-(-x + y) + x + Double y) + -(-x + y) + z;
    thus -(y + z) = -(-(-k + -(y + z)) + -(k + -(y + z))) by Def5
      .= -(z + -(k + -(y + z))) by A4
      .= -(-(-(-(-x + y) + x + Double y) + -(-x + y) + -(y + z) + z) + z) by
LATTICES:def 5;
  end;
A6: now
    let x, y, z, u;
    set k = -(-(-(-x + y) + x + Double y) + -(-x + y) + -(y + z) + z) + z;
    thus u = -(-(-k + u) + -(k + u)) by Def5
      .= -(-(-(y + z) + u) + -(k + u)) by A5;
  end;
A7: now
    let x, y, z, v;
    set k = -(-(Double v + v) + v);
    set l = -(-(Double v + v) + v) + Double v;
    set v5 = Double v + Double v + v;
A8: -(Double v + v + l) = -(-(-(Double v + v + l) + -(Double v + v) + l) +
    l) by A1
      .= -(-(-(Double v + v + l) + l + -(Double v + v)) + l) by LATTICES:def 5;
    thus k = -(-(-(v + Double v) + k) + -((-(-(-(-(Double v + v) + v) + (
Double v + v) + Double v) + -(-(Double v + v) + v) + -(v + Double v) + Double v
    ) + Double v) + k)) by A6
      .= -(-(-(v + Double v) + k) + -((-(-(Double v + v + k + Double v) + k
    + Double v + -(v + Double v)) + Double v) + k)) by LATTICES:def 5
      .= -(-(-(v + Double v) + k) + -((-(-((Double v + v) + k + Double v) +
    (k + Double v) + -(v + Double v)) + Double v) + k)) by LATTICES:def 5
      .= -(-(-(v + Double v) + k) + -((-(-(Double v + v + (k + Double v)) +
    l + -(v + Double v)) + Double v) + k)) by LATTICES:def 5
      .= -(-(-(v + Double v) + k) + -(-(-(Double v + v + l) + l + -(v +
    Double v)) + l)) by LATTICES:def 5
      .= -(-(-(Double v + v) + k) + -(Double v + v + Double v + k)) by A8,
LATTICES:def 5
      .= -(-(-(Double v + v) + k) + -(v5 + k)) by LATTICES:def 5;
  end;
A9: now
    let x;
    set k = -(-(Double x + x) + x) + -(Double x + x);
    set l = -(-(-(Double x + x) + x) + (Double x + Double x + x));
A10: -(Double x + x) = -(-(-(-(-(Double x + x) + x) + (Double x + x) +
Double x) + -(-(Double x + x) + x) + -(Double x + x)) + -(x + -(Double x + x)))
    by A4
      .= -(-(-(-(-(Double x + x) + x) + (Double x + x) + Double x) + k) + -(
    x + -(Double x + x))) by LATTICES:def 5
      .= -(-(-(-(-(Double x + x) + x) + (Double x + x + Double x)) + k) + -(
    x + -(Double x + x))) by LATTICES:def 5
      .= -(-(l + k) + -(x + -(Double x + x))) by LATTICES:def 5;
    l = -(-(-k + l) + -(k + l)) by Def5
      .= -(-(-(Double x + x) + x) + -(k + l)) by A7;
    hence -(-(-(Double x + x) + x) + (Double x + Double x + x)) = -(Double x +
    x) by A10;
  end;
A11: now
    let x;
A12: -(-(Double x + x) + x) = -(-(-(-(Double x + x) + x) + (Double x + x)
    + x) + x) by A2
      .= -(-(-(-(Double x + x) + x) + (Double x + x + x)) + x) by
LATTICES:def 5
      .= -(-(-(-(Double x + x) + x) + (Double x + Double x)) + x) by
LATTICES:def 5;
    thus x = -(-(-(-(-(Double x + x) + x) + (Double x + Double x)) + x) + -(-(
    -(Double x + x) + x) + (Double x + Double x) + x)) by Def5
      .= -(-(-(-(-(Double x + x) + x) + (Double x + Double x)) + x) + -(-(-(
    Double x + x) + x) + ((Double x + Double x) + x))) by LATTICES:def 5
      .= -(-(-(Double x + x) + x) + -(Double x + x)) by A9,A12;
  end;
A13: now
    let x, y;
    thus y = -(-(-(-(-(Double x + x) + x) + -(Double x + x)) + y) + -(-(-(
    Double x + x) + x) + -(Double x + x) + y)) by Def5
      .= -(-(-(-(Double x + x) + x) + -(Double x + x) + y) + -(x + y)) by A11;
  end;
A14: now
    let x;
    thus -(-(Double x + x) + x) + Double x = -(-(-(Double x + x) + (-(-(Double
x + x) + x) + Double x)) + -((Double x + x) + (-(-(Double x + x) + x) + Double
    x))) by Def5
      .= -(-(-(Double x + x) + (-(-(Double x + x) + x) + Double x)) + -((-(-
    (Double x + x) + x) + ((Double x + x) + Double x)))) by LATTICES:def 5
      .= -(-(-(Double x + x) + (-(-(Double x + x) + x) + Double x)) + -((-(-
    (Double x + x) + x) + (Double x + Double x + x)))) by LATTICES:def 5
      .= -(-(-(Double x + x) + (-(-(Double x + x) + x) + Double x)) + -(
    Double x + x)) by A9
      .= -(-(-(-(Double x + x) + x) + -(Double x + x) + Double x) + -(Double
    x + x)) by LATTICES:def 5;
  end;
A15: now
    let x, y;
    thus Double x = -(-(-(-(Double x + x) + x) + -(Double x + x) + Double x )
    + -(x + Double x)) by A13
      .= -(-(Double x + x) + x) + Double x by A14;
  end;
  set x = the Element of G;
  set c = Double x, d = -(-(Double x + x) + x);
  take d, c;
  thus thesis by A15;
end;
