 reserve L for non empty LattStr;
 reserve v100,v102,v2,v1,v0,v3,v101 for Element of L;

theorem
  (for v0 holds v0"/\"v0 = v0) &
  (for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
  (for v0 holds v0"\/"v0 = v0) &
  (for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
  (for v2,v1,v0 holds ((v0"\/"v1)"/\"(v2"\/"v1))"/\"v1 = v1) &
  (for v2,v1,v0 holds ((v0"/\"v1)"\/"(v2"/\"v1))"\/"v1 = v1) &
  (for v1,v2,v0 holds v0"/\"(v1"\/"(v0"\/"v2)) = v0) implies
    for v0,v1,v2 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)
  proof
    assume
A2: for v0 holds v0"/\"v0 = v0;
    assume
A3: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
    assume
A4: for v0 holds v0"\/"v0 = v0;
    assume
A5: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
    assume
A6: for v2,v1,v0 holds ((v0"\/"v1)"/\"(v2"\/"v1))"/\"v1 = v1;
A8: for v2,v1,v0 holds v1"/\"((v0"\/"v1)"/\"(v2"\/"v1)) = v1
    proof let v2,v1,v0;
      ((v0"\/"v1)"/\"(v2"\/"v1))"/\"v1 =
        v1"/\"((v0"\/"v1)"/\"(v2"\/"v1)) by A3;
      hence thesis by A6;
    end;
    assume
A10:for v2,v1,v0 holds ((v0"/\"v1)"\/"(v2"/\"v1))"\/"v1 = v1;
A12:for v2,v1,v0 holds v1"\/"((v0"/\"v1)"\/"(v2"/\"v1)) = v1
    proof let v2,v1,v0;
      ((v0"/\"v1)"\/"(v2"/\"v1))"\/"v1 =
         v1"\/"((v0"/\"v1)"\/"(v2"/\"v1)) by A5;
      hence thesis by A10;
    end;
    assume
A14:for v1,v2,v0 holds v0"/\"(v1"\/"(v0"\/"v2)) = v0;
A20:for v100,v102 holds v100"/\"(v102"\/"v100) = v100
    proof let v100,v102;
      (v102"\/"v100)"/\"(v102"\/"v100) = v102"\/"v100 by A2;
      hence thesis by A8;
    end;
A23:for v2,v1,v0 holds v0"\/"((v0"/\"v1)"\/"(v2"/\"v0)) = v0
    proof let v2,v1,v0;
      v1"/\"v0 = v0"/\"v1 by A3;
      hence thesis by A12;
    end;
A25:for v2,v0,v1 holds v0"\/"((v1"/\"v0)"\/"(v0"/\"v2)) = v0
    proof let v2,v0,v1;
      v2"/\"v0 = v0"/\"v2 by A3;
      hence thesis by A12;
    end;
A28:for v100,v102 holds v100"\/"(v102"/\"v100) = v100
    proof let v100,v102;
      (v102"/\"v100)"\/"(v102"/\"v100) = v102"/\"v100 by A4;
      hence thesis by A12;
    end;
A32:for v102,v2,v101,v1 holds
     ((v1"/\"v101)"\/"(v2"/\"v101))"/\"(v101"/\"
      (v102"\/"((v1"/\"v101)"\/"(v2"/\"v101)))) =
       (v1"/\"v101)"\/"(v2"/\"v101)
    proof let v102,v2,v101,v1;
      v101"\/"((v1"/\"v101)"\/"(v2"/\"v101)) = v101 by A12;
      hence thesis by A8;
    end;
A35:for v1,v0,v2 holds v0"/\"(v1"\/"(v2"\/"v0)) = v0
    proof let v1,v0,v2;
      v0"\/"v2 = v2"\/"v0 by A5;
      hence thesis by A14;
    end;
A38:for v102,v101,v1 holds
     (v1"\/"v101)"\/"(v101"\/"(v102"/\"(v1"\/"v101))) = v1"\/"v101
    proof let v102,v101,v1;
      v101"/\"(v1"\/"v101) = v101 by A20;
      hence thesis by A12;
    end;
A41:for v1,v0 holds v0"\/"(v0"/\"v1) = v0
    proof let v1,v0;
      v1"/\"v0 = v0"/\"v1 by A3;
      hence thesis by A28;
    end;
A44:for v102,v1,v101 holds
     (v101"/\"v1)"/\"(v101"/\"(v102"\/"(v101"/\"v1))) = v101"/\"v1
    proof let v102,v1,v101;
      v101"\/"(v101"/\"v1) = v101 by A41;
      hence thesis by A8;
    end;
A47:for v1,v0,v2 holds v0"/\"((v2"\/"v0)"\/"v1) = v0
    proof let v1,v0,v2;
      v1"\/"(v2"\/"v0) = (v2"\/"v0)"\/"v1 by A5;
      hence thesis by A35;
    end;
A51:for v101,v102,v1 holds (v1"/\"v102)"/\"(v101"\/"v102) = v1"/\"v102
    proof let v101,v102,v1;
      v102"\/"(v1"/\"v102) = v102 by A28;
      hence thesis by A35;
    end;
A55:for v102,v101,v1 holds (v1"/\"v101)"/\"(v101"\/"v102) = v1"/\"v101
    proof let v102,v101,v1;
      v101"\/"(v1"/\"v101) = v101 by A28;
      hence thesis by A47;
    end;
A58:for v2,v1,v0 holds v0"\/"((v0"/\"v1)"\/"(v0"/\"v2)) = v0
    proof let v2,v1,v0;
      v2"/\"v0 = v0"/\"v2 by A3;
      hence thesis by A23;
    end;
A61:for v102,v2,v101,v1 holds
     ((v1"/\"v101)"\/"(v101"/\"v2))"/\"(v101"/\"(v102"\/"
       ((v1"/\"v101)"\/"(v101"/\"v2)))) = (v1"/\"v101)"\/"(v101"/\"v2)
    proof let v102,v2,v101,v1;
      v101"\/"((v1"/\"v101)"\/"(v101"/\"v2)) = v101 by A25;
      hence thesis by A8;
    end;
A65:for v0,v1,v2 holds (v2"\/"v1)"\/"(v0"/\"v1) = v2"\/"v1
    proof let v0,v1,v2;
      (v0"/\"v1)"/\"(v2"\/"v1) = v0"/\"v1 by A51;
      hence thesis by A28;
    end;
A69:for v100,v1,v102 holds
     (v100"/\"(v102"/\"v1))"/\"v102 = v100"/\"(v102"/\"v1)
    proof let v100,v1,v102;
      v102"\/"(v102"/\"v1) = v102 by A41;
      hence thesis by A51;
    end;
A72:for v0,v2,v1 holds v1"/\"(v0"/\"(v1"/\"v2)) = v0"/\"(v1"/\"v2)
    proof let v0,v2,v1;
      (v0"/\"(v1"/\"v2))"/\"v1 = v1"/\"(v0"/\"(v1"/\"v2)) by A3;
      hence thesis by A69;
    end;
  A76: for v101,v102 holds
   ((v102"/\"v101)"\/"(v102"/\"v101))"/\"(v101"/\"v102) =
      (v102"/\"v101)"\/"(v102"/\"v101)
  proof let v101,v102;
    v102"\/"((v102"/\"v101)"\/"(v102"/\"v101)) = v102 by A58;
    hence thesis by A32;
  end;

  A79: for v1,v0 holds (v0"/\"v1)"/\"(v1"/\"v0) = (v0"/\"v1)"\/"(v0"/\"v1)
  proof let v1,v0;
    (v0"/\"v1)"\/"(v0"/\"v1) = v0"/\"v1 by A4;
    hence thesis by A76;
  end;

  A81: for v1,v0 holds (v0"/\"v1)"/\"(v1"/\"v0) = v0"/\"v1
  proof let v1,v0;
    (v0"/\"v1)"\/"(v0"/\"v1) = v0"/\"v1 by A4;
    hence thesis by A79;
  end;

  A84: for v102,v100,v1 holds
   v100"\/"(v102"/\"(v1"/\"v100)) = v100"\/"(v1"/\"v100)
  proof let v102,v100,v1;
    v100"\/"(v1"/\"v100) = v100 by A28;
    hence thesis by A65;
  end;

  A87: for v1,v0,v2 holds v0"\/"(v1"/\"(v2"/\"v0)) = v0
  proof let v1,v0,v2;
    v0"\/"(v2"/\"v0) = v0 by A28;
    hence thesis by A84;
  end;

  A90: for v102,v1,v100 holds
    v100"\/"(v102"/\"(v100"/\"v1)) = v100"\/"(v100"/\"v1)
  proof let v102,v1,v100;
    v100"\/"(v100"/\"v1) = v100 by A41;
    hence thesis by A65;
  end;

  A93: for v1,v2,v0 holds v0"\/"(v1"/\"(v0"/\"v2)) = v0
  proof let v1,v2,v0;
    v0"\/"(v0"/\"v2) = v0 by A41;
    hence thesis by A90;
  end;

  A95: for v1,v0,v2 holds v0"\/"((v2"/\"v0)"/\"v1) = v0
  proof let v1,v0,v2;
    v1"/\"(v2"/\"v0) = (v2"/\"v0)"/\"v1 by A3;
    hence thesis by A87;
  end;

  A99: for v102,v2,v100,v1 holds
    v100"\/"(((v1"/\"v100)"/\"v2)"\/"
      (v102"/\"(v100"\/"((v1"/\"v100)"/\"v2)))) =
        v100"\/"((v1"/\"v100)"/\"v2)
  proof let v102,v2,v100,v1;
    v100"\/"((v1"/\"v100)"/\"v2) = v100 by A95;
    hence thesis by A38;
  end;

  A102: for v3,v2,v0,v1 holds
    v0"\/"(((v1"/\"v0)"/\"v2)"\/"(v3"/\"v0)) = v0"\/"((v1"/\"v0)"/\"v2)
  proof let v3,v2,v0,v1;
    v0"\/"((v1"/\"v0)"/\"v2) = v0 by A95;
    hence thesis by A99;
  end;

  A104: for v3,v2,v0,v1 holds v0"\/"(((v1"/\"v0)"/\"v2)"\/"(v3"/\"v0)) = v0
  proof let v3,v2,v0,v1;
    v0"\/"((v1"/\"v0)"/\"v2) = v0 by A95;
    hence thesis by A102;
  end;

  A107: for v102,v0,v1 holds
   (v0"/\"v1)"/\"((v1"/\"v0)"\/"v102) = (v0"/\"v1)"/\"(v1"/\"v0)
  proof let v102,v0,v1;
    (v0"/\"v1)"/\"(v1"/\"v0) = v0"/\"v1 by A81;
    hence thesis by A55;
  end;

  A110: for v2,v0,v1 holds (v0"/\"v1)"/\"((v1"/\"v0)"\/"v2) = v0"/\"v1
  proof let v2,v0,v1;
    (v0"/\"v1)"/\"(v1"/\"v0) = v0"/\"v1 by A81;
    hence thesis by A107;
  end;

  A113: for v100,v101,v0 holds
   (v100"/\"v101)"/\"(v100"/\"(v0"\/"v101)) = v100"/\"v101
  proof let v100,v101,v0;
    (v0"\/"v101)"\/"(v100"/\"v101) = v0"\/"v101 by A65;
    hence thesis by A44;
  end;

  A117: for v100,v2,v102 holds
   (v100"/\"(v102"/\"v2))"/\"(v100"/\"v102) = v100"/\"(v102"/\"v2)
  proof let v100,v2,v102;
    v102"\/"(v100"/\"(v102"/\"v2)) = v102 by A93;
    hence thesis by A44;
  end;

  A120: for v0,v2,v1 holds (v0"/\"v1)"/\"(v0"/\"(v1"/\"v2)) = v0"/\"(v1"/\"v2)
  proof let v0,v2,v1;
    (v0"/\"(v1"/\"v2))"/\"(v0"/\"v1) = (v0"/\"v1)"/\"(v0"/\"(v1"/\"v2)) by A3;
    hence thesis by A117;
  end;

  A123: for v101,v1,v2,v100 holds
   v100"\/"((v100"/\"v101)"\/"(v1"/\"(v100"/\"v2))) = v100
  proof let v101,v1,v2,v100;
    v100"/\"(v1"/\"(v100"/\"v2)) = v1"/\"(v100"/\"v2) by A72;
    hence thesis by A58;
  end;

  A127: for v101,v102,v1 holds
   (((v1"/\"v102)"/\"v101)"\/"(v101"/\"v102))"/\"(v101"/\"v102) =
      ((v1"/\"v102)"/\"v101)"\/"(v101"/\"v102)
  proof let v101,v102,v1;
    v102"\/"(((v1"/\"v102)"/\"v101)"\/"(v101"/\"v102)) = v102 by A104;
    hence thesis by A61;
  end;

  A130: for v2,v1,v0 holds
   (v2"/\"v1)"/\"(((v0"/\"v1)"/\"v2)"\/"(v2"/\"v1)) =
      ((v0"/\"v1)"/\"v2)"\/"(v2"/\"v1)
  proof let v2,v1,v0;
    (((v0"/\"v1)"/\"v2)"\/"(v2"/\"v1))"/\"(v2"/\"v1) =
      (v2"/\"v1)"/\"(((v0"/\"v1)"/\"v2)"\/"(v2"/\"v1)) by A3;
    hence thesis by A127;
  end;

  A133: for v0,v1,v2 holds v0"/\"v1 = ((v2"/\"v1)"/\"v0)"\/"(v0"/\"v1)
  proof let v0,v1,v2;
    (v0"/\"v1)"/\"(((v2"/\"v1)"/\"v0)"\/"(v0"/\"v1)) = v0"/\"v1 by A20;
    hence thesis by A130;
  end;

  A138: for v101,v3,v100 holds
   ((v100"/\"v101)"\/"(v101"/\"(v100"/\"v3)))"/\"(v101"/\"v100) =
      (v100"/\"v101)"\/"(v101"/\"(v100"/\"v3))
  proof let v101,v3,v100;
    v100"\/"((v100"/\"v101)"\/"(v101"/\"(v100"/\"v3))) = v100 by A123;
    hence thesis by A61;
  end;

  A141: for v1,v2,v0 holds
   (v1"/\"v0)"/\"((v0"/\"v1)"\/"(v1"/\"(v0"/\"v2))) =
     (v0"/\"v1)"\/"(v1"/\"(v0"/\"v2))
  proof let v1,v2,v0;
    ((v0"/\"v1)"\/"(v1"/\"(v0"/\"v2)))"/\"(v1"/\"v0) =
      (v1"/\"v0)"/\"((v0"/\"v1)"\/"(v1"/\"(v0"/\"v2))) by A3;
    hence thesis by A138;
  end;

  A144: for v0,v2,v1 holds v0"/\"v1 = (v1"/\"v0)"\/"(v0"/\"(v1"/\"v2))
  proof let v0,v2,v1;
    (v0"/\"v1)"/\"((v1"/\"v0)"\/"(v0"/\"(v1"/\"v2))) = v0"/\"v1 by A110;
    hence thesis by A141;
  end;

  A149: for v100,v101,v2 holds
   (v100"/\"v101)"\/"((v100"/\"(v2"\/"v101))"/\"v101) =
     (v100"/\"(v2"\/"v101))"/\"v101
  proof let v100,v101,v2;
    (v100"/\"v101)"/\"(v100"/\"(v2"\/"v101)) = v100"/\"v101 by A113;
    hence thesis by A133;
  end;

  A152: for v0,v1,v2 holds
   (v0"/\"v1)"\/"(v1"/\"(v0"/\"(v2"\/"v1))) = (v0"/\"(v2"\/"v1))"/\"v1
  proof let v0,v1,v2;
    (v0"/\"(v2"\/"v1))"/\"v1 = v1"/\"(v0"/\"(v2"\/"v1)) by A3;
    hence thesis by A149;
  end;

  A154: for v0,v1,v2 holds v1"/\"v0 = (v0"/\"(v2"\/"v1))"/\"v1
  proof let v0,v1,v2;
    (v0"/\"v1)"\/"(v1"/\"(v0"/\"(v2"\/"v1))) = v1"/\"v0 by A144;
    hence thesis by A152;
  end;

  A157: for v1,v0,v2 holds v0"/\"v1 = v0"/\"(v1"/\"(v2"\/"v0))
  proof let v1,v0,v2;
    (v1"/\"(v2"\/"v0))"/\"v0 = v0"/\"(v1"/\"(v2"\/"v0)) by A3;
    hence thesis by A154;
  end;

  A160: for v1,v0,v2 holds v0"/\"((v2"\/"v0)"/\"v1) = v0"/\"v1
  proof let v1,v0,v2;
    v1"/\"(v2"\/"v0) = (v2"\/"v0)"/\"v1 by A3;
    hence thesis by A157;
  end;

  A164: for v102,v101,v1 holds
   (v1"/\"v101)"/\"(v101"/\"v102) = (v1"/\"v101)"/\"v102
  proof let v102,v101,v1;
    v101"\/"(v1"/\"v101) = v101 by A28;
    hence thesis by A160;
  end;

  A168: for v102,v1,v101 holds
   (v101"/\"v1)"/\"(v101"/\"v102) = (v101"/\"v1)"/\"v102
  proof let v102,v1,v101;
    v101"\/"(v101"/\"v1) = v101 by A41;
    hence thesis by A160;
  end;

  A171: for v2,v1,v0 holds (v0"/\"v1)"/\"(v1"/\"v2) = v0"/\"(v1"/\"v2)
  proof let v2,v1,v0;
    (v0"/\"v1)"/\"(v0"/\"(v1"/\"v2)) = (v0"/\"v1)"/\"(v1"/\"v2) by A168;
    hence thesis by A120;
  end;

  A173: for v2,v1,v0 holds (v0"/\"v1)"/\"v2 = v0"/\"(v1"/\"v2)
  proof let v2,v1,v0;
    (v0"/\"v1)"/\"(v1"/\"v2) = (v0"/\"v1)"/\"v2 by A164;
    hence thesis by A171;
  end;

  A175: for v1,v2,v0 holds v0"/\"(v1"/\"(v0"/\"v2)) = (v0"/\"v1)"/\"v2
  proof let v1,v2,v0;
    (v0"/\"v1)"/\"(v0"/\"v2) = v0"/\"(v1"/\"(v0"/\"v2)) by A173;
    hence thesis by A168;
  end;

  A177: for v1,v2,v0 holds v1"/\"(v0"/\"v2) = (v0"/\"v1)"/\"v2
  proof let v1,v2,v0;
    v0"/\"(v1"/\"(v0"/\"v2)) = v1"/\"(v0"/\"v2) by A72;
    hence thesis by A175;
  end;

  A180: for v0,v2,v1 holds v0"/\"(v1"/\"v2) = v1"/\"(v0"/\"v2)
  proof let v0,v2,v1;
    (v1"/\"v0)"/\"v2 = v1"/\"(v0"/\"v2) by A173;
    hence thesis by A177;
  end;
  let v0,v2,v1;
  v1"/\"(v0"/\"v2) = v0"/\"(v1"/\"v2) by A180; then
  v1"/\"(v0"/\"v2) = v0"/\"(v2"/\"v1) by A3;
  hence thesis by A3;
end;
