reserve L for non empty LattStr;
reserve v3,v101,v100,v102,v103,v2,v1,v0 for Element of L;

theorem Lemma1: :: QLT-1
   (for v1,v0 holds v0"/\"v1 = v1"/\"v0) & :: /\-commutative
   (for v0,v2,v1 holds ::: link law
      (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
   (for v0 holds v0"\/"v0 = v0) & :: \/-idempotent
   (for v2,v1,v0 holds
      (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) & :: \/-associative
   (for v1,v0 holds v0"\/"v1 = v1"\/"v0) & :: \/-commutative
   (for v0,v2,v1 holds  :: link law
      (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
   (for v1,v2,v0 holds :: QLT-distributivity
       v0"/\"(v1"\/"(v0"/\"v2)) = v0"/\"(v1"\/"v2)) implies
   for v1,v2,v3 holds :: distributivity
       (v1"/\"v2)"\/"(v1"/\"v3) = v1"/\"(v2"\/"v3)
proof
assume A1: for v1,v0 holds v0"/\"v1 = v1"/\"v0;
assume A2: for v0,v2,v1 holds
 (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2);
assume A3: for v0 holds v0"\/"v0 = v0;
assume A4: for v2,v1,v0 holds
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2);
assume A5: for v1,v0 holds v0"\/"v1 = v1"\/"v0;
assume A6: for v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2);

A8: 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 A1;
  hence thesis by A6;
end;

assume A9: for v1,v2,v0 holds
  v0"/\"(v1"\/"(v0"/\"v2)) = v0"/\"(v1"\/"v2);

A12: 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 A2;
end;

A15: for v102,v100 holds (v100"\/"v102)"\/"v102 = v100"\/"v102
proof let v102,v100;
  v102"\/"v102 = v102 by A3;
  hence thesis by A4;
end;

A18: for v1,v0 holds v1"\/"(v0"\/"v1) = v0"\/"v1
proof let v1,v0;
  (v0"\/"v1)"\/"v1 = v1"\/"(v0"\/"v1) by A5;
  hence thesis by A15;
end;

A21: for v2,v0,v1 holds (v1"\/"v0)"\/"v2 = v0"\/"(v1"\/"v2)
proof let v2,v0,v1;
  v0"\/"v1 = v1"\/"v0 by A5;
  hence thesis by A4;
end;

A24: for v0,v2,v1 holds
  v0"\/"(v1"\/"v2) = v1"\/"(v0"\/"v2)
proof let v0,v2,v1;
  (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2) by A4;
  hence thesis by A21;
end;

A27: for v102,v101 holds
 v101"/\"(v101"\/"(v101"/\"v102)) = v101"\/"(v101"/\"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A3;
  hence thesis by A8;
end;

A30: for v1,v0 holds v0"/\"(v0"\/"v1) = v0"\/"(v0"/\"v1)
proof let v1,v0;
  v0"/\"(v0"\/"(v0"/\"v1)) = v0"/\"(v0"\/"v1) by A9;
  hence thesis by A27;
end;

A33: for v1,v0,v2 holds
  v0"/\"(v1"\/"(v2"/\"v0)) = v0"/\"(v1"\/"v2)
proof let v1,v0,v2;
  v0"/\"v2 = v2"/\"v0 by A1;
  hence thesis by A9;
end;

A35: for v0,v1,v2 holds
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"v1)) = v0"/\"(v1"\/"v2)
proof let v0,v1,v2;
  v1"\/"v2 = v2"\/"v1 by A5;
  hence thesis by A12;
end;

A37: for v0,v1 holds v0"\/"(v1"/\"v0) = v0"/\"(v0"\/"v1)
proof let v0,v1;
  v0"/\"v1 = v1"/\"v0 by A1;
  hence thesis by A30;
end;

A40: for v1,v2,v101 holds
 (v1"\/"(v101"/\"v2))"\/"(v101"/\"(v1"\/"v2)) =
    (v1"\/"(v101"/\"v2))"/\"((v1"\/"(v101"/\"v2))"\/"v101)
proof let v1,v2,v101;
  v101"/\"(v1"\/"(v101"/\"v2)) = v101"/\"(v1"\/"v2) by A9;
  hence thesis by A37;
end;

A43: for v1,v2,v0 holds
 (v1"/\"(v0"\/"v2))"\/"(v0"\/"(v1"/\"v2)) =
   (v0"\/"(v1"/\"v2))"/\"((v0"\/"(v1"/\"v2))"\/"v1)
proof let v1,v2,v0;
  (v0"\/"(v1"/\"v2))"\/"(v1"/\"(v0"\/"v2)) =
    (v1"/\"(v0"\/"v2))"\/"(v0"\/"(v1"/\"v2)) by A5;
  hence thesis by A40;
end;

A46: for v0,v2,v1 holds
 v1"\/"((v0"/\"(v1"\/"v2))"\/"(v0"/\"v2)) =
  (v1"\/"(v0"/\"v2))"/\"((v1"\/"(v0"/\"v2))"\/"v0)
proof let v0,v2,v1;
  (v0"/\"(v1"\/"v2))"\/"(v1"\/"(v0"/\"v2)) =
    v1"\/"((v0"/\"(v1"\/"v2))"\/"(v0"/\"v2)) by A24;
  hence thesis by A43;
end;

A49: for v1,v2,v0 holds
 v0"\/"((v1"/\"v2)"\/"(v1"/\"(v0"\/"v2))) =
   (v0"\/"(v1"/\"v2))"/\"((v0"\/"(v1"/\"v2))"\/"v1)
proof let v1,v2,v0;
  (v1"/\"(v0"\/"v2))"\/"(v1"/\"v2) =
    (v1"/\"v2)"\/"(v1"/\"(v0"\/"v2)) by A5;
  hence thesis by A46;
end;

A51: for v1,v0,v2 holds
 v0"\/"(v1"/\"(v2"\/"v0)) =
  (v0"\/"(v1"/\"v2))"/\"((v0"\/"(v1"/\"v2))"\/"v1)
proof let v1,v0,v2;
  (v1"/\"v2)"\/"(v1"/\"(v0"\/"v2)) = v1"/\"(v2"\/"v0) by A35;
  hence thesis by A49;
end;

A53: for v1,v0,v2 holds
 v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"(v1"/\"v2))"/\"(v1"\/"(v0"\/"(v1"/\"v2)))
proof let v1,v0,v2;
  (v0"\/"(v1"/\"v2))"\/"v1 = v1"\/"(v0"\/"(v1"/\"v2)) by A5;
  hence thesis by A51;
end;

A56: for v101,v0,v2,v1 holds
 (v0"\/"(v1"/\"v2))"/\"(v101"\/"(v0"\/"(v1"/\"v2))) =
   (v0"\/"(v1"/\"v2))"/\"(v101"\/"(v0"\/"v1))
proof let v101,v0,v2,v1;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A8;
  hence thesis by A33;
end;

A59: for v1,v0,v2 holds
 v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"(v1"/\"v2))"/\"(v1"\/"(v0"\/"v1))
proof let v1,v0,v2;
  (v0"\/"(v1"/\"v2))"/\"(v1"\/"(v0"\/"(v1"/\"v2))) =
     (v0"\/"(v1"/\"v2))"/\"(v1"\/"(v0"\/"v1)) by A56;
  hence thesis by A53;
end;

A61: for v1,v0,v2 holds
 v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1)
proof let v1,v0,v2;
  v1"\/"(v0"\/"v1) = v0"\/"v1 by A18;
  hence thesis by A59;
end;

A63: for v1,v0,v2 holds
  v0"\/"(v1"/\"(v2"\/"v0)) = (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2))
proof let v1,v0,v2;
  (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) =
    (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) by A1;
  hence thesis by A61;
end;

A65: for v1,v0,v2 holds
  v0"\/"(v1"/\"(v2"\/"v0)) = v0"\/"(v1"/\"v2)
proof let v1,v0,v2;
  (v0"\/"v1)"/\"(v0"\/"(v1"/\"v2)) = v0"\/"(v1"/\"v2) by A8;
  hence thesis by A63;
end;

A68: for v101,v2,v102 holds
 (v101"/\"v2)"\/"(v101"/\"(v102"\/"v2)) =
    (v101"/\"v2)"\/"(v101"/\"v102)
proof let v101,v2,v102;
  v101"/\"(v102"\/"(v101"/\"v2)) = v101"/\"(v102"\/"v2) by A9;
  hence thesis by A65;
end;

for v0,v2,v1 holds
  v0"/\"(v1"\/"v2) = (v0"/\"v1)"\/"(v0"/\"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"(v2"\/"v1)) = v0"/\"(v1"\/"v2) by A35;
  hence thesis by A68;
end;
hence thesis;
end;
