theorem Th19:
  (for v holds (J,v |= CQC_Sub(S) iff J,v.Val_S(v,S) |= S))
  implies for v holds (J,v |= CQC_Sub(Sub_not S) iff J,v.Val_S(v,Sub_not S) |=
  Sub_not S)
proof
  assume
A1: for v holds (J,v |= CQC_Sub(S) iff J,v.Val_S(v,S) |= S);
  let v;
A2: J,v |= 'not' CQC_Sub(S) iff not J,v |= CQC_Sub(S) by VALUAT_1:17;
  not J,v.Val_S(v,S) |= S iff J,v.Val_S(v,S) |= Sub_not S by Th17;
  hence thesis by A1,A2,Th18,SUBSTUT1:29;
end;
