theorem Th84:
  dom @RestrictSub(x,p,Sub) misses dom ((@Sub)|RSub1(p)) \/ dom ((
  @Sub)|RSub2(p,Sub))
proof
  set X = {y : y in still_not-bound_in p & y is Element of dom Sub & y <> x &
  y <> Sub.y};
A1: dom ((@Sub)|RSub2(p,Sub)) = dom @Sub /\ RSub2(p,Sub) by RELAT_1:61;
  RestrictSub(x,p,Sub) = Sub|X by SUBSTUT1:def 6;
  then RestrictSub(x,p,Sub) = (@Sub|X) by SUBSTUT1:def 2;
  then dom @RestrictSub(x,p,Sub) = dom (@Sub|X) by SUBSTUT1:def 2;
  then
A2: dom @RestrictSub(x,p,Sub) = dom @Sub /\ X by RELAT_1:61;
A3: dom ((@Sub)|RSub1(p)) = dom @Sub /\ RSub1(p) by RELAT_1:61;
  now
    assume dom @RestrictSub(x,p,Sub) meets dom ((@Sub)|RSub1(p)) \/ dom ((@
    Sub)|RSub2(p,Sub));
    then consider b being object such that
A4: b in dom @RestrictSub(x,p,Sub) and
A5: b in dom ((@Sub)|RSub1(p)) \/ dom ((@Sub)|RSub2(p,Sub)) by XBOOLE_0:3;
    b in X by A2,A4,XBOOLE_0:def 4;
    then
A6: ex y st b = y & y in still_not-bound_in p & y is Element of dom Sub &
    y <> x & y <> Sub.y;
A7: now
      assume b in dom ((@Sub)|RSub2(p,Sub));
      then b in RSub2(p,Sub) by A1,XBOOLE_0:def 4;
      then ex y1 st y1 = b & y1 in still_not-bound_in p & y1 = (@ Sub).y1 by
Def10;
      hence contradiction by A6,SUBSTUT1:def 2;
    end;
    now
      assume b in dom ((@Sub)|RSub1(p));
      then b in RSub1(p) by A3,XBOOLE_0:def 4;
      then ex y1 st y1 = b & not y1 in still_not-bound_in p by Def9;
      hence contradiction by A6;
    end;
    hence contradiction by A5,A7,XBOOLE_0:def 3;
  end;
  hence thesis;
end;
