reserve Al for QC-alphabet;
reserve PHI for Consistent Subset of CQC-WFF(Al),
        p,q,r,s for Element of CQC-WFF(Al),
        A for non empty set,
        J for interpretation of Al,A,
        v for Element of Valuations_in(Al,A),
        m,n,i,j,k for Nat,
        l for CQC-variable_list of k,Al,
        P for QC-pred_symbol of k,Al,
        x,y,z for bound_QC-variable of Al,
        b for QC-symbol of Al,
        PR for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve Al2 for Al-expanding QC-alphabet,
        J2 for interpretation of Al2,A,
        Jp for interpretation of Al,A,
        v2 for Element of Valuations_in(Al2,A),
        vp for Element of Valuations_in(Al,A);

theorem Th8:
  for p,r,x,P,l for Al2 being Al-expanding QC-alphabet
  holds Al2-Cast(VERUM(Al)) = VERUM(Al2) &
  Al2-Cast(P!l) = Al2-Cast(P)!Al2-Cast(l) &
  Al2-Cast('not' p) = 'not' (Al2-Cast(p)) &
  Al2-Cast(p '&' r) = (Al2-Cast(p)) '&' (Al2-Cast(r)) &
  Al2-Cast(All(x,p)) = All(Al2-Cast(x),Al2-Cast(p))
proof
  let p,r,x,P,l;
  let Al2 be Al-expanding QC-alphabet;
A1: the_arity_of P = len l by Th1;
A2: the_arity_of Al2-Cast(P) = len (Al2-Cast(l)) by Th1;
  thus Al2-Cast(VERUM(Al)) = VERUM(Al2);
  thus Al2-Cast(P!l) = <*P*>^l by A1,QC_LANG1:def 12
           .= Al2-Cast(P)!Al2-Cast(l) by A2,QC_LANG1:def 12;
  thus Al2-Cast('not' p) = 'not' (Al2-Cast(p));
  thus Al2-Cast(p '&' r) = (Al2-Cast(p)) '&' (Al2-Cast(r));
  thus Al2-Cast(All(x,p)) = All(Al2-Cast(x),Al2-Cast(p));
end;
