theorem Th82:
  dom ((@Sub)|RSub1(p)) misses dom ((@Sub)|RSub2(p,Sub))
proof
  now
    assume dom ((@Sub)|RSub1(p)) meets dom ((@Sub)|RSub2(p,Sub));
    then consider a being object such that
A1: a in dom ((@Sub)|RSub1(p)) /\ dom ((@Sub)|RSub2(p,Sub)) by XBOOLE_0:4;
    dom ((@Sub)|RSub1(p)) = dom (@Sub) /\ RSub1(p) & dom ((@Sub)|RSub2(p,
    Sub)) = (dom (@Sub) /\ RSub2(p,Sub)) by RELAT_1:61;
    then a in (dom (@Sub) /\ (dom (@Sub) /\ RSub1(p))) /\ RSub2(p,Sub) by A1,
XBOOLE_1:16;
    then a in dom (@Sub) /\ dom (@Sub) /\ RSub1(p) /\ RSub2(p,Sub) by
XBOOLE_1:16;
    then a in dom (@Sub) /\ (RSub1(p) /\ RSub2(p,Sub)) by XBOOLE_1:16;
    then
A2: a in RSub1(p) /\ RSub2(p,Sub) by XBOOLE_0:def 4;
    then a in RSub2(p,Sub) by XBOOLE_0:def 4;
    then
A3: ex b being bound_QC-variable of Al st b = a & b in still_not-bound_in p & b
    = (@Sub).b by Def10;
    a in RSub1(p) by A2,XBOOLE_0:def 4;
    then ex b being bound_QC-variable of Al st b = a & not b in
     still_not-bound_in p by Def9;
    hence contradiction by A3;
  end;
  hence thesis;
end;
