theorem Th82:
  x in H1 /\ H2 iff x in H1 & x in H2
proof
  thus x in H1 /\ H2 implies x in H1 & x in H2
  proof
    assume x in H1 /\ H2;
    then x in carr(H1) /\ carr(H2) by Def10;
    hence thesis by XBOOLE_0:def 4;
  end;
  assume x in H1 & x in H2;
  then x in carr(H1) /\ carr(H2);
  hence thesis by Def10;
end;
