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;

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;
