
theorem min1:
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 & s in S2 by A,XBOOLE_0:def 4;
   then -s in -S1 & -s in -S2;
   hence o in (-S1) /\ (-S2) by A,XBOOLE_0:def 4;
   end;
now let o be object;
  assume o in (-S1) /\ (-S2);
  then A: o in -S1 & o in -S2 by XBOOLE_0:def 4;
  then consider s1 being Element of L such that B: -s1 = o & s1 in S1;
  consider s2 being Element of L such that C: -s2 = o & s2 in S2 by A;
  s1 = -(-s2) by B,C .= s2;
  then s1 in S1 /\ S2 by B,C,XBOOLE_0:def 4;
  hence o in -(S1 /\ S2) by B;
  end;
hence thesis by A,TARSKI:2;
end;
