theorem Th15:
  for v holds (J,v |= CQC_Sub(Sub_P(P,ll,Sub)) iff J,v.Val_S(v,
  Sub_P(P,ll,Sub)) |= Sub_P(P,ll,Sub))
proof
  set S9 = Sub_P(P,ll,Sub);
  set ll9 = CQC_Subst(ll,Sub);
  reconsider p = P!ll9 as Element of CQC-WFF(Al);
  reconsider ll9 as CQC-variable_list of k,Al;
  let v;
A1: Valid(p,J).v = TRUE iff v*'ll9 in J.P by VALUAT_1:7;
A2: (v.Val_S(v,S9))*'ll in J.P iff Valid(P!ll,J).((v.Val_S(v,S9))) = TRUE by
VALUAT_1:7;
A3: J,v.Val_S(v,S9) |= P!ll iff J,v.Val_S(v,S9) |= Sub_P(P,ll,Sub)`1 by Th14;
  J,v |= CQC_Sub(Sub_P(P,ll,Sub)) iff J,v |= p by Th8;
  hence thesis by A1,A2,A3,Th13,VALUAT_1:def 7;
end;
