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 Th3:
  S is Al-Sub_VERUM implies CQC_Sub(S) = VERUM(Al)
proof
  ex F being Function of QC-Sub-WFF(Al),QC-WFF(Al) st CQC_Sub(S) = F.S & for S9
being Element of QC-Sub-WFF(Al) holds (S9 is Al-Sub_VERUM implies
 F.S9 = VERUM(Al)) & ( S9 is Sub_atomic implies F.S9 = (the_pred_symbol_of
 ((S9)`1))! CQC_Subst(Sub_the_arguments_of S9,(S9)`2)) &
  (S9 is Sub_negative implies F.S9 = 'not' (F.
  (Sub_the_argument_of S9))) & (S9 is Sub_conjunctive implies F.S9 = (F.
  Sub_the_left_argument_of S9) '&' (F.Sub_the_right_argument_of S9)) & (S9 is
  Sub_universal implies F.S9 = Quant(S9,F.Sub_the_scope_of S9)) by
SUBSTUT1:def 38;
  hence thesis;
end;
