reserve A for QC-alphabet;
reserve i,j,k,l,m,n for Nat;
reserve a,b,e for set;
reserve t,u,v,w,z for QC-symbol of A;
reserve p,q,r,s for Element of CQC-WFF(A);
reserve x for Element of bound_QC-variables(A);
reserve ll for CQC-variable_list of k,A;
reserve P for QC-pred_symbol of k,A;
reserve f,h for Element of Funcs(bound_QC-variables(A),bound_QC-variables(A)),
  K,L for Element of Fin bound_QC-variables(A);

theorem Th35:
  for q,t,K,f holds [q,t,K,f] in SepQuadruples p implies q is_subformula_of p
proof
  defpred P[Element of CQC-WFF(A),set,set,set] means $1 is_subformula_of p;
A1: now
    let q,t,K,f such that
    ['not' q,t,K,f] in SepQuadruples p;
    q is_subformula_of 'not' q by Th10;
    hence P['not' q,t,K,f] implies P[q,t,K,f] by QC_LANG2:57;
  end;
A2: now
    let q,x,t,K,f such that
    [All(x,q),t,K,f] in SepQuadruples p;
    q is_subformula_of All(x,q) by Th12;
    hence P[All(x,q),t,K,f] implies P[q,t++,K \/ {x},f+*(x .--> x.t)] by
QC_LANG2:57;
  end;
A3: now
    let q,r,t,K,f such that
    [q '&' r, t, K,f] in SepQuadruples p;
A4: r is_subformula_of q '&'r by Th11;
    q is_subformula_of q '&'r by Th11;
    hence P[q '&' r,t,K,f] implies P[q,t,K,f] & P[r,t+QuantNbr(q),K,f] by A4,
QC_LANG2:57;
  end;
A5: P[p,index p,{}.bound_QC-variables(A),id bound_QC-variables(A)];
  thus for q,t,K,f st [q,t,K,f] in SepQuadruples p holds P[q,t,K,f] from
  Sepregression(A5,A1,A3,A2);
end;
