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 Th11:
  for f being FinSequence of CQC-WFF(Al2),g being FinSequence of CQC-WFF(Al)
  st f=g holds Ant f = Ant g & Suc f = Suc g
proof
 let f be FinSequence of CQC-WFF(Al2),g be FinSequence of CQC-WFF(Al) such that
A1: f = g;
  per cases;
  suppose
A2: len f > 0;
    then consider k being Nat such that
A3:  len f = k + 1 by NAT_1:6;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    thus Ant f = g|(Seg k) by A1,A2,A3,CALCUL_1:def 1
              .= Ant g by A1,A2,A3,CALCUL_1:def 1;
    Suc f = g.(len g) by A1,A2,CALCUL_1:def 2
         .= Suc g by A1,A2,CALCUL_1:def 2;
    hence thesis;
  end;
  suppose
A4:  not len f > 0;
    thus Ant f = {} by A4,CALCUL_1:def 1
              .= Ant g by A1,A4,CALCUL_1:def 1;
    thus Suc f = VERUM(Al2) by A4, CALCUL_1:def 2
              .= VERUM(Al)
              .= Suc g by A1,A4, CALCUL_1:def 2;
  end;
end;
