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 Th12:
  x in dom S`2 implies (v.Val_S(v,S)).x = Val_S(v,S).x
proof
  assume x in dom S`2;
  then
A1: x in dom @S`2 by SUBSTUT1:def 2;
  rng @S`2 c= bound_QC-variables(Al) & dom v = bound_QC-variables(Al)
   by FUNCT_2:def 1;
  then x in dom Val_S(v,S) by A1,RELAT_1:27;
  hence thesis by FUNCT_4:13;
end;
