theorem
  for S, v holds (J,v |= CQC_Sub(S) iff J,v.Val_S(v,S) |= S)
proof
  defpred Pro[Element of CQC-Sub-WFF(Al)] means for v holds (J,v |= CQC_Sub($1)
  iff J,v.Val_S(v,$1) |= $1);
A1: for S,S9 being Element of CQC-Sub-WFF(Al), x being bound_QC-variable of Al,
  SQ be second_Q_comp of [S,x], k being Nat,ll being
  CQC-variable_list of k,Al, P being (QC-pred_symbol of k,Al), e being
  Element of vSUB(Al) holds Pro[Sub_P(P,ll,e)] & (S is Al-Sub_VERUM
  implies Pro[S]) & (Pro[S] implies Pro[Sub_not S]) & (S`2 = (S9)`2 & Pro[S]
  & Pro[S9] implies Pro[CQCSub_&(S,S9)]) & ([S,x] is quantifiable &
  Pro[S] implies Pro[CQCSub_All([S,x], SQ)]) by Th4,Th15,Th19,Th25,Th88;
  thus for S holds Pro[S] from SubCQCInd1(A1);
end;
