reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;
reserve a for Element of A;

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;
