theorem Th78:
  (for y st y in dom vS1 holds not y in still_not-bound_in All(x,p
  )) implies for y st y in (dom vS1) \ {x} holds not y in still_not-bound_in p
proof
  assume
A1: for y st y in dom vS1 holds not y in still_not-bound_in All(x,p);
  let y such that
A2: y in (dom vS1) \ {x};
  (dom vS1) \ {x} c= dom vS1 by XBOOLE_1:36;
  then not y in still_not-bound_in All(x,p) by A1,A2;
  then
A3: not y in (still_not-bound_in p) \ {x} by QC_LANG3:12;
A4: {x} \/ ((still_not-bound_in p) \ {x}) = {x} \/ still_not-bound_in p by
XBOOLE_1:39;
  not y in {x} by A2,XBOOLE_0:def 5;
  then not y in {x} \/ ((still_not-bound_in p) \ {x}) by A3,XBOOLE_0:def 3;
  hence thesis by A4,XBOOLE_0:def 3;
end;
