theorem Th4:
  S is Al-Sub_VERUM implies for v holds (J,v |= CQC_Sub(S) iff J,v.
  Val_S(v,S) |= S)
proof
  assume
A1: S is Al-Sub_VERUM;
  let v;
  ex Sub st S = [VERUM(Al),Sub] by A1,SUBSTUT1:def 19;
  then S`1 = VERUM(Al);
  then J,v.Val_S(v,S) |= VERUM(Al) iff J,v.Val_S(v,S) |= S;
  hence J,v |= CQC_Sub(S) implies J,v.Val_S(v,S) |= S by VALUAT_1:32;
  J,v.Val_S(v,S) |= S implies J,v |= VERUM(Al) by VALUAT_1:32;
  hence thesis by A1,Th3;
end;
