 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,v0,v1 holds v0"/\"((v1"\/"v0)"/\"(v0"\/"v2)) = v0
  proof let v2,v0,v1;
    v2"\/"v0 = v0"\/"v2 by A5;
    hence thesis by A8;
  end;

  A25: 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;

  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,v100 holds v100"/\"(v100"\/"v102) = v100
  proof let v102,v100;
    (v100"\/"v102)"\/"(v100"\/"v102) = v100"\/"v102 by A4;
    hence thesis by A14;
  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;

  A37: for v1,v2,v0 holds v0"/\"((v0"\/"v2)"\/"v1) = v0
  proof let v1,v2,v0;
    v1"\/"(v0"\/"v2) = (v0"\/"v2)"\/"v1 by A5;
    hence thesis by A14;
  end;

  A41: 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;

  A45: for v102,v101,v1 holds
   (v1"/\"v101)"/\"(v101"/\"(v102"\/"(v1"/\"v101))) = v1"/\"v101
  proof let v102,v101,v1;
    v101"\/"(v1"/\"v101) = v101 by A28;
    hence thesis by A8;
  end;

  A49: for v1,v2,v101 holds
    (v1"\/"(v101"\/"v2))"\/"v101 = v1"\/"(v101"\/"v2)
  proof let v1,v2,v101;
    v101"/\"(v1"\/"(v101"\/"v2)) = v101 by A14;
    hence thesis by A28;
  end;

  A52: 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 A5;
    hence thesis by A49;
  end;

  A56: for v102,v1,v101 holds
   (v101"\/"v1)"\/"(v101"\/"(v102"/\"(v101"\/"v1))) = v101"\/"v1
  proof let v102,v1,v101;
    v101"/\"(v101"\/"v1) = v101 by A32;
    hence thesis by A12;
  end;

  A60: for v1,v101 holds (v101"\/"v1)"\/"v101 = v101"\/"v1
  proof let v1,v101;
    v101"/\"(v101"\/"v1) = v101 by A32;
    hence thesis by A28;
  end;

  A63: for v1,v0 holds v0"\/"(v0"\/"v1) = v0"\/"v1
  proof let v1,v0;
    (v0"\/"v1)"\/"v0 = v0"\/"(v0"\/"v1) by A5;
    hence thesis by A60;
  end;

  A65: 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;

  A69: 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 A23;
    hence thesis by A12;
  end;

  A73: 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 A65;
  end;

  A77: for v101,v1,v2,v102 holds
   (v1"\/"(v102"\/"v2))"\/"(((v1"\/"(v102"\/"v2))"/\"v101)"\/"v102) =
      v1"\/"(v102"\/"v2)
  proof let v101,v1,v2,v102;
    v102"/\"(v1"\/"(v102"\/"v2)) = v102 by A14;
    hence thesis by A25;
  end;

  A80: for v3,v0,v2,v1 holds
   (v0"\/"(v1"\/"v2))"\/"(v1"\/"((v0"\/"(v1"\/"v2))"/\"v3)) = v0"\/"(v1"\/"v2)
  proof let v3,v0,v2,v1;
    ((v0"\/"(v1"\/"v2))"/\"v3)"\/"v1 = v1"\/"((v0"\/"(v1"\/"v2))"/\"v3) by A5;
    hence thesis by A77;
  end;

  A83: for v101,v2,v1,v102 holds
   ((v102"\/"v1)"\/"v2)"\/"((((v102"\/"v1)"\/"v2)"/\"v101)"\/"v102) =
     (v102"\/"v1)"\/"v2
  proof let v101,v2,v1,v102;
    v102"/\"((v102"\/"v1)"\/"v2) = v102 by A37;
    hence thesis by A25;
  end;

  A86: for v3,v2,v1,v0 holds
   ((v0"\/"v1)"\/"v2)"\/"(v0"\/"(((v0"\/"v1)"\/"v2)"/\"v3)) = (v0"\/"v1)"\/"v2
  proof let v3,v2,v1,v0;
    (((v0"\/"v1)"\/"v2)"/\"v3)"\/"v0 = v0"\/"(((v0"\/"v1)"\/"v2)"/\"v3) by A5;
    hence thesis by A83;
  end;

  A89: for v101,v100,v0 holds
   (v100"\/"v101)"\/"(v101"\/"(v0"/\"v100)) = v100"\/"v101
  proof let v101,v100,v0;
    (v0"/\"v100)"/\"(v100"\/"v101) = v0"/\"v100 by A73;
    hence thesis by A41;
  end;

  A93: for v102,v2,v1,v100 holds
   v100"/\"(((v100"\/"v1)"\/"v2)"/\"
     (v102"\/"(v100"/\"((v100"\/"v1)"\/"v2)))) =
       v100"/\"((v100"\/"v1)"\/"v2)
  proof let v102,v2,v1,v100;
    v100"/\"((v100"\/"v1)"\/"v2) = v100 by A37;
    hence thesis by A45;
  end;

  A96: for v3,v2,v1,v0 holds
   v0"/\"(((v0"\/"v1)"\/"v2)"/\"(v3"\/"v0)) = v0"/\"((v0"\/"v1)"\/"v2)
  proof let v3,v2,v1,v0;
    v0"/\"((v0"\/"v1)"\/"v2) = v0 by A37;
    hence thesis by A93;
  end;

  A98: for v3,v2,v1,v0 holds v0"/\"(((v0"\/"v1)"\/"v2)"/\"(v3"\/"v0)) = v0
  proof let v3,v2,v1,v0;
    v0"/\"((v0"\/"v1)"\/"v2) = v0 by A37;
    hence thesis by A96;
  end;

  A101: for v100,v2,v102 holds
   (v100"\/"(v102"\/"v2))"\/"(v100"\/"v102) = v100"\/"(v102"\/"v2)
  proof let v100,v2,v102;
    v102"/\"(v100"\/"(v102"\/"v2)) = v102 by A14;
    hence thesis by A56;
  end;

  A104: 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 A5;
    hence thesis by A101;
  end;

  A106: for v3,v0,v2,v1 holds
   (v0"\/"(v2"\/"v1))"\/"(v1"\/"((v0"\/"(v1"\/"v2))"/\"v3)) = v0"\/"(v1"\/"v2)
  proof let v3,v0,v2,v1;
    v1"\/"v2 = v2"\/"v1 by A5;
    hence thesis by A80;
  end;

  A110: for v101,v1,v102 holds
   (((v102"\/"v1)"\/"v101)"/\"(v101"\/"v102))"\/"(v101"\/"v102) =
      ((v102"\/"v1)"\/"v101)"/\"(v101"\/"v102)
  proof let v101,v1,v102;
    v102"/\"(((v102"\/"v1)"\/"v101)"/\"(v101"\/"v102)) = v102 by A98;
    hence thesis by A69;
  end;

  A113: for v2,v1,v0 holds
   (v2"\/"v0)"\/"(((v0"\/"v1)"\/"v2)"/\"(v2"\/"v0)) =
      ((v0"\/"v1)"\/"v2)"/\"(v2"\/"v0)
  proof let v2,v1,v0;
    (((v0"\/"v1)"\/"v2)"/\"(v2"\/"v0))"\/"(v2"\/"v0) =
      (v2"\/"v0)"\/"(((v0"\/"v1)"\/"v2)"/\"(v2"\/"v0)) by A5;
    hence thesis by A110;
  end;

  A116: for v0,v2,v1 holds v0"\/"v1 = ((v1"\/"v2)"\/"v0)"/\"(v0"\/"v1)
  proof let v0,v2,v1;
    (v0"\/"v1)"\/"(((v1"\/"v2)"\/"v0)"/\"(v0"\/"v1)) = v0"\/"v1 by A28;
    hence thesis by A113;
  end;

  A120: for v3,v2,v0,v1 holds
   ((v0"\/"v1)"\/"v2)"\/"(v0"\/"(((v1"\/"v0)"\/"v2)"/\"v3)) =
      (v0"\/"v1)"\/"v2
  proof let v3,v2,v0,v1;
    v0"\/"v1 = v1"\/"v0 by A5;
    hence thesis by A86;
  end;

  A122: for v2,v1,v0 holds (v2"\/"(v0"\/"v1))"/\"(v2"\/"v0) = v2"\/"v0
  proof let v2,v1,v0;
    (v0"\/"v1)"\/"v2 = v2"\/"(v0"\/"v1) by A5;
    hence thesis by A116;
  end;

  A125: for v0,v2,v1 holds (v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)) = v0"\/"v1
  proof let v0,v2,v1;
    (v0"\/"(v1"\/"v2))"/\"(v0"\/"v1) = (v0"\/"v1)"/\"(v0"\/"(v1"\/"v2)) by A3;
    hence thesis by A122;
  end;

  A128: for v101,v100,v2 holds
   (v100"\/"v101)"/\"((v101"\/"(v2"/\"v100))"\/"v100) =
     (v101"\/"(v2"/\"v100))"\/"v100
  proof let v101,v100,v2;
    (v100"\/"v101)"\/"(v101"\/"(v2"/\"v100)) = v100"\/"v101 by A89;
    hence thesis by A116;
  end;

  A131: for v1,v0,v2 holds
   (v0"\/"v1)"/\"(v0"\/"(v1"\/"(v2"/\"v0))) = (v1"\/"(v2"/\"v0))"\/"v0
  proof let v1,v0,v2;
    (v1"\/"(v2"/\"v0))"\/"v0 = v0"\/"(v1"\/"(v2"/\"v0)) by A5;
    hence thesis by A128;
  end;

  A133: for v1,v0,v2 holds v0"\/"v1 = (v1"\/"(v2"/\"v0))"\/"v0
  proof let v1,v0,v2;
    (v0"\/"v1)"/\"(v0"\/"(v1"\/"(v2"/\"v0))) = v0"\/"v1 by A125;
    hence thesis by A131;
  end;

  A135: 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 A5;
    hence thesis by A133;
  end;

  A138: for v1,v0,v2 holds v0"\/"((v2"/\"v0)"\/"v1) = v0"\/"v1
  proof let v1,v0,v2;
    v1"\/"(v2"/\"v0) = (v2"/\"v0)"\/"v1 by A5;
    hence thesis by A135;
  end;

  A142: for v102,v101,v1 holds
   (v1"\/"v101)"\/"(v101"\/"v102) = (v1"\/"v101)"\/"v102
  proof let v102,v101,v1;
    v101"/\"(v1"\/"v101) = v101 by A20;
    hence thesis by A138;
  end;

  A146: for v102,v1,v101 holds
   (v101"\/"v1)"\/"(v101"\/"v102) = (v101"\/"v1)"\/"v102
  proof let v102,v1,v101;
    v101"/\"(v101"\/"v1) = v101 by A32;
    hence thesis by A138;
  end;

  A149: 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 A146;
    hence thesis by A104;
  end;

  A151: for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)
  proof let v2,v1,v0;
    (v0"\/"v1)"\/"(v1"\/"v2) = (v0"\/"v1)"\/"v2 by A142;
    hence thesis by A149;
  end;

  A153: for v3,v0,v1,v2 holds
    v0"\/"((v1"\/"v2)"\/"(v2"\/"((v0"\/"(v2"\/"v1))"/\"v3))) =
      v0"\/"(v2"\/"v1)
  proof let v3,v0,v1,v2;
    (v0"\/"(v1"\/"v2))"\/"(v2"\/"((v0"\/"(v2"\/"v1))"/\"v3)) =
       v0"\/"((v1"\/"v2)"\/"(v2"\/"((v0"\/"(v2"\/"v1))"/\"v3))) by A151;
    hence thesis by A106;
  end;

  A155: for v3,v0,v1,v2 holds
    v0"\/"(v1"\/"(v2"\/"(v2"\/"((v0"\/"(v2"\/"v1))"/\"v3)))) = v0"\/"(v2"\/"v1)
  proof let v3,v0,v1,v2;
    (v1"\/"v2)"\/"(v2"\/"((v0"\/"(v2"\/"v1))"/\"v3)) =
      v1"\/"(v2"\/"(v2"\/"((v0"\/"(v2"\/"v1))"/\"v3))) by A151;
    hence thesis by A153;
  end;

  A157: for v3,v0,v1,v2 holds
   v0"\/"(v1"\/"(v2"\/"((v0"\/"(v2"\/"v1))"/\"v3))) = v0"\/"(v2"\/"v1)
  proof let v3,v0,v1,v2;
    v2"\/"(v2"\/"((v0"\/"(v2"\/"v1))"/\"v3)) =
       v2"\/"((v0"\/"(v2"\/"v1))"/\"v3) by A63;
    hence thesis by A155;
  end;

  A159: for v3,v2,v0,v1 holds
   (v0"\/"(v1"\/"v2))"\/"(v0"\/"(((v1"\/"v0)"\/"v2)"/\"v3)) = (v0"\/"v1)"\/"v2
  proof let v3,v2,v0,v1;
    (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A151;
    hence thesis by A120;
  end;

  A161: for v3,v1,v2,v0 holds
    (v0"\/"(v1"\/"v2))"\/"(v0"\/"((v1"\/"(v0"\/"v2))"/\"v3)) = (v0"\/"v1)"\/"v2
  proof let v3,v1,v2,v0;
    (v1"\/"v0)"\/"v2 = v1"\/"(v0"\/"v2) by A151;
    hence thesis by A159;
  end;

  A163: for v3,v1,v2,v0 holds
    v0"\/"((v1"\/"v2)"\/"(v0"\/"((v1"\/"(v0"\/"v2))"/\"v3))) = (v0"\/"v1)"\/"v2
  proof let v3,v1,v2,v0;
    (v0"\/"(v1"\/"v2))"\/"(v0"\/"((v1"\/"(v0"\/"v2))"/\"v3)) =
      v0"\/"((v1"\/"v2)"\/"(v0"\/"((v1"\/"(v0"\/"v2))"/\"v3))) by A151;
    hence thesis by A161;
  end;

  A165: for v3,v1,v2,v0 holds
    v0"\/"(v1"\/"(v2"\/"(v0"\/"((v1"\/"(v0"\/"v2))"/\"v3)))) = (v0"\/"v1)"\/"v2
  proof let v3,v1,v2,v0;
    (v1"\/"v2)"\/"(v0"\/"((v1"\/"(v0"\/"v2))"/\"v3)) =
      v1"\/"(v2"\/"(v0"\/"((v1"\/"(v0"\/"v2))"/\"v3))) by A151;
    hence thesis by A163;
  end;

  A167: for v1,v2,v0 holds v0"\/"(v1"\/"(v0"\/"v2)) = (v0"\/"v1)"\/"v2
  proof
    now let v3,v1,v2,v0;
      v1"\/"(v2"\/"(v0"\/"((v1"\/"(v0"\/"v2))"/\"v3))) = v1"\/"(v0"\/"v2)
        by A157;
      hence v0"\/"(v1"\/"(v0"\/"v2)) = (v0"\/"v1)"\/"v2 by A165;
    end;
    hence thesis;
  end;

  A169: for v1,v2,v0 holds v1"\/"(v0"\/"v2) = (v0"\/"v1)"\/"v2
  proof let v1,v2,v0;
    v0"\/"(v1"\/"(v0"\/"v2)) = v1"\/"(v0"\/"v2) by A52;
    hence thesis by A167;
  end;

  A172: for v0,v2,v1 holds v0"\/"(v1"\/"v2) = v1"\/"(v0"\/"v2)
  proof let v0,v2,v1;
    (v1"\/"v0)"\/"v2 = v1"\/"(v0"\/"v2) by A151;
    hence thesis by A169;
  end;
  let v0,v2,v1;
  v1"\/"(v0"\/"v2) = v0"\/"(v1"\/"v2) by A172; then
  (v0"\/"v2)"\/"v1 = v0"\/"(v1"\/"v2) by A5;
  hence thesis by A5;
end;
