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 Th8:
  CQC_Sub(Sub_P(P,ll,Sub)) = P!CQC_Subst(ll,Sub)
proof
A1: P!ll is atomic by QC_LANG1:def 18;
A2: Sub_P(P,ll,Sub) = [P!ll,Sub] by SUBSTUT1:9;
  then
A3: Sub_P(P,ll,Sub)`2 = Sub;
  Sub_P(P,ll,Sub)`1 = P!ll by A2;
  then
  Sub_the_arguments_of Sub_P(P,ll,Sub) = ll & the_pred_symbol_of (Sub_P(P,
  ll, Sub)`1) = P by A1,QC_LANG1:def 22,SUBSTUT1:def 29;
  hence thesis by A3,Th6;
end;
