
theorem
for L being add-associative right_zeroed right_complementable
            non empty addLoopStr
for S1,S2 being Subset of L holds -(S1 \/ S2) = (-S1) \/ (-S2)
proof
let L be add-associative right_zeroed right_complementable
         non empty addLoopStr, S1,S2 be Subset of L;
A: now let o be object;
   assume o in -(S1 \/ S2);
   then consider s being Element of L such that A: -s = o & s in S1 \/ S2;
   s in S1 or s in S2 by A,XBOOLE_0:def 3;
   then -s in -S1 or -s in -S2;
   hence o in (-S1) \/ (-S2) by A,XBOOLE_0:def 3;
   end;
now let o be object;
  assume A: o in (-S1) \/ (-S2);
  per cases by A,XBOOLE_0:def 3;
  suppose o in -S1;
    then consider s1 being Element of L such that B: -s1 = o & s1 in S1;
    s1 in S1 \/ S2 by B,XBOOLE_0:def 3;
    hence o in -(S1 \/ S2) by B;
    end;
  suppose o in -S2;
    then consider s1 being Element of L such that B: -s1 = o & s1 in S2;
    s1 in S1 \/ S2 by B,XBOOLE_0:def 3;
    hence o in -(S1 \/ S2) by B;
    end;
  end;
hence thesis by A,TARSKI:2;
end;
