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;
