reserve A for QC-alphabet;
reserve a,b,b1,b2,c,d for object,
  i,j,k,n for Nat,
  x,y,x1,x2 for bound_QC-variable of A,
  P for QC-pred_symbol of k,A,
  ll for CQC-variable_list of k,A,
  l1 ,l2 for FinSequence of QC-variables(A),
  p for QC-formula of A,
  s,t for QC-symbol of A;
reserve Sub for CQC_Substitution of A;
reserve finSub for finite CQC_Substitution of A;
reserve e for Element of vSUB(A);
reserve S,S9,S1,S2,S19,S29,T1,T2 for Element of QC-Sub-WFF(A);
reserve B for Element of [:QC-Sub-WFF(A),bound_QC-variables(A):];
reserve SQ for second_Q_comp of B;

theorem Th25:
  (S is A-Sub_VERUM implies ((@S`1).1)`1 = 0) & (S is Sub_atomic
  implies ex k being Nat st (@S`1).1 is QC-pred_symbol of k,A)
& (S is
Sub_negative implies ((@S`1).1)`1 = 1) & (S is Sub_conjunctive implies ((@S`1).
  1)`1 = 2) & (S is Sub_universal implies ((@S`1).1)`1 = 3)
proof
  thus S is A-Sub_VERUM implies ((@S`1).1)`1 = 0;
  thus S is Sub_atomic implies ex k being Nat st (@S`1).1 is
  QC-pred_symbol of k,A
  proof
    assume S is Sub_atomic;
    then consider
    k being Nat, P being QC-pred_symbol of k,A, ll being
    QC-variable_list of k,A, e being Element of vSUB(A) such that
A1: S = Sub_P(P,ll,e);
    S = [P!ll,e] by A1,Th9;
    then S`1 = P!ll;
    then @S`1 = <*P*>^ll by QC_LANG1:8;
    then (@S`1).1 = P by FINSEQ_1:41;
    hence thesis;
  end;
  thus S is Sub_negative implies ((@S`1).1)`1 = 1
  proof
    assume S is Sub_negative;
    then consider S9 such that
A2: S = Sub_not S9;
    S`1 = 'not' (S9)`1 by A2;
    then @(S`1).1 = [1,0] by FINSEQ_1:41;
    hence thesis;
  end;
  thus S is Sub_conjunctive implies ((@S`1).1)`1 = 2
  proof
    assume S is Sub_conjunctive;
    then consider S1,S2 such that
A3: S = Sub_&(S1,S2) & S1`2 = S2`2;
    S = [(S1`1) '&' (S2`1),S1`2] by A3,Def21;
    then S`1 = (S1`1) '&' (S2`1);
    then @S`1 = <*[2,0]*>^(@S1`1^@S2`1) by FINSEQ_1:32;
    then @(S`1).1 = [2,0] by FINSEQ_1:41;
    hence thesis;
  end;
  thus S is Sub_universal implies ((@S`1).1)`1 = 3
  proof
    assume S is Sub_universal;
    then consider B,SQ such that
A4: S = Sub_All(B,SQ) & B is quantifiable;
    S = [All(B`2,(B`1)`1),SQ] by A4,Def24;
    then S`1 = All(B`2,(B`1)`1);
    then @S`1 = <*[3,0]*>^(<*B`2*>^@(B`1)`1) by FINSEQ_1:32;
    then (@S`1).1 = [3,0] by FINSEQ_1:41;
    hence thesis;
  end;
end;
