
theorem lempart0:
for L being add-associative right_zeroed right_complementable
   non empty addLoopStr holds -{0.L} = {0.L}
proof
let L be add-associative right_zeroed right_complementable
         non empty addLoopStr;
A: -{0.L} c= {0.L}
   proof
   now let o be object;
     assume o in -{0.L}; then
     consider x being Element of L such that
     B: o = -x & x in {0.L};
     x = 0.L by B,TARSKI:def 1;
     hence o in {0.L} by B;
     end;
   hence thesis;
   end;
{0.L} c= -{0.L}
   proof
   now let o be object;
     assume B: o in {0.L}; then
     reconsider x = o as Element of L;
     C: x = 0.L by B,TARSKI:def 1;
     -x in -{0.L} by B;
     hence o in -{0.L} by C;
     end;
   hence thesis;
   end;
hence thesis by A;
end;
