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

theorem QLTMod2:
(for v1,v0 holds v0"/\"v1 = v1"/\"v0) &
(for v0,v2,v1 holds (v0"/\"(v1"\/"v2))"\/"(v0"/\"v1) = v0"/\"(v1"\/"v2)) &
(for v0 holds v0"\/"v0 = v0) &
(for v2,v1,v0 holds (v0"\/"v1)"\/"v2 = v0"\/"(v1"\/"v2)) &
(for v1,v0 holds v0"\/"v1 = v1"\/"v0) &
(for v0,v2,v1 holds (v0"\/"(v1"/\"v2))"/\"(v0"\/"v1) = v0"\/"(v1"/\"v2)) &
(for v2,v1,v0 holds (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2)))
implies
for v1,v2,v3 st v1"\/"v2 = v2 holds v1"\/"(v3"/\"v2) = (v1"\/"v3)"/\"v2
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 v2,v1,v0 holds
 (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2));

assume not thesis; then
consider c1, c2, c3 being Element of L such that
A10: c1"\/"c2 = c2 and
A11:  c1"\/"(c3"/\"c2) <> (c1"\/"c3)"/\"c2;

c1"\/"(c2"/\"c3) <> (c1"\/"c3)"/\"c2 by A1,A11; then
A15:  c1"\/"(c2"/\"c3) <> c2"/\"(c1"\/"c3) by A1;

A17: 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;

A19: for v0,v2,v1 holds v0"/\"(v1"\/"(v0"/\"(v1"\/"v2))) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  (v0"/\"v1)"\/"(v0"/\"(v1"\/"v2)) = v0"/\"(v1"\/"(v0"/\"(v1"\/"v2))) by A9;
  hence thesis by A17;
end;

A22: for v102,v101 holds v101"\/"v102 = v101"\/"(v101"\/"v102)
proof let v102,v101;
  v101"\/"v101 = v101 by A3;
  hence thesis by A4;
end;

A27: for v1,v2,v0 holds v0"/\"(v1"\/"(v0"/\"v2)) = (v0"/\"v2)"\/"(v0"/\"v1)
proof let v1,v2,v0;
  (v0"/\"v1)"\/"(v0"/\"v2) = v0"/\"(v1"\/"(v0"/\"v2)) by A9;
  hence thesis by A5;
end;

A29: for v1,v2,v0 holds v0"/\"(v1"\/"(v0"/\"v2)) = v0"/\"(v2"\/"(v0"/\"v1))
proof let v1,v2,v0;
  (v0"/\"v2)"\/"(v0"/\"v1) = v0"/\"(v2"\/"(v0"/\"v1)) by A9;
  hence thesis by A27;
end;

A35: for v102 holds c1"\/"(c2"/\"(c1"\/"v102)) = c2"/\"(c1"\/"v102)
proof let v102;
  c2"/\"(c1"\/"(c2"/\"(c1"\/"v102))) = c1"\/"(c2"/\"(c1"\/"v102)) by A10,A8;
  hence thesis by A19;
end;

A38: for v0 holds c1"\/"(c2"/\"(v0"\/"c1)) = c2"/\"(c1"\/"v0)
proof let v0;
  c1"\/"v0 = v0"\/"c1 by A5;
  hence thesis by A35;
end;

A40: for v0,v2,v1 holds v0"/\"((v1"\/"v2)"\/"(v0"/\"v1)) = v0"/\"(v1"\/"v2)
proof let v0,v2,v1;
  v0"/\"(v1"\/"(v0"/\"(v1"\/"v2))) = v0"/\"((v1"\/"v2)"\/"(v0"/\"v1)) by A29;
  hence thesis by A19;
end;

A42: for v2,v1,v0 holds v0"/\"(v1"\/"(v2"\/"(v0"/\"v1))) = v0"/\"(v1"\/"v2)
proof let v2,v1,v0;
  (v1"\/"v2)"\/"(v0"/\"v1) = v1"\/"(v2"\/"(v0"/\"v1)) by A4;
  hence thesis by A40;
end;

A44: for v0 holds c2"/\"(v0"\/"(c2"/\"c1)) = c1"\/"(c2"/\"v0)
proof let v0;
  c2"/\"(c1"\/"(c2"/\"v0)) = c2"/\"(v0"\/"(c2"/\"c1)) by A29;
  hence thesis by A10,A8;
end;

A45: c2"/\"c1 = c1"/\"c2 by A1;

A50: for v101 holds
 c2"/\"(v101"\/"(c1"\/"(c2"/\"v101))) = c2"/\"(v101"\/"(c1"/\"c2))
proof let v101;
  c1"\/"(c2"/\"v101) = c2"/\"(v101"\/"(c1"/\"c2)) by A45,A44;
  hence thesis by A19;
end;

A53: for v0 holds c2"/\"(v0"\/"c1) = c2"/\"(v0"\/"(c1"/\"c2))
proof let v0;
  c2"/\"(v0"\/"(c1"\/"(c2"/\"v0))) = c2"/\"(v0"\/"c1) by A42;
  hence thesis by A50;
end;

A57: for v0 holds c1"\/"(c2"/\"(v0"\/"(c1"/\"c2))) = c1"\/"(c2"/\"v0)
proof let v0;
  c1"\/"(c2"/\"v0) = c2"/\"(v0"\/"(c1"/\"c2)) by A45,A44;
  hence thesis by A22;
end;

A59: for v0 holds c1"\/"(c2"/\"(v0"\/"c1)) = c1"\/"(c2"/\"v0)
proof let v0;
  c2"/\"(v0"\/"(c1"/\"c2)) = c2"/\"(v0"\/"c1) by A53;
  hence thesis by A57;
end;

for v0 holds c2"/\"(c1"\/"v0) = c1"\/"(c2"/\"v0)
proof let v0;
  c1"\/"(c2"/\"(v0"\/"c1)) = c2"/\"(c1"\/"v0) by A38;
  hence thesis by A59;
end;
hence thesis by A15;
end;
