theorem Th62:
  (for v,w holds v|still_not-bound_in p = w|still_not-bound_in p
  implies (J,v |= p iff J,w |= p)) & (for v,w holds v|still_not-bound_in q = w|
still_not-bound_in q implies (J,v |= q iff J,w |= q)) implies for v,w holds v|
still_not-bound_in p '&' q = w|still_not-bound_in p '&' q implies (J,v |= p '&'
  q iff J,w |= p '&' q)
proof
  assume
A1: ( for v,w holds v|still_not-bound_in p = w|still_not-bound_in p
  implies ( J,v |= p iff J,w |= p))& for v,w holds v|still_not-bound_in q = w|
  still_not-bound_in q implies (J,v |= q iff J,w |= q);
  set X = (still_not-bound_in p) \/ (still_not-bound_in q);
  let v,w;
A2: still_not-bound_in p '&' q = X by QC_LANG3:10;
  assume v|still_not-bound_in p '&' q = w|still_not-bound_in p '&' q;
  then
  v|still_not-bound_in p = w|still_not-bound_in p & v|still_not-bound_in q
  = w |still_not-bound_in q by A2,RELAT_1:153,XBOOLE_1:7;
  then J,v |= p & J,v |= q iff J,w |= p & J,w |= q by A1;
  hence thesis by VALUAT_1:18;
end;
