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 Th38:
  for q,t,K,f st [q,t,K,f] in SepQuadruples p holds
  still_not-bound_in q c= still_not-bound_in p \/ K
proof
  deffunc f(QC-formula of A) = still_not-bound_in $1;
  defpred P[QC-formula of A,set, set, set] means f($1) c= f(p) \/ $3;
A1: for q,t,K,f st ['not' q,t,K,f] in SepQuadruples p & P['not' q,t,K,f]
  holds P[q,t,K,f] by QC_LANG3:7;
A2: now
    let q,r,t,K,f such that
    [q '&' r, t, K,f] in SepQuadruples p and
A3: P[q '&' r,t,K,f];
A4: still_not-bound_in q '&' r = still_not-bound_in q \/
    still_not-bound_in r by QC_LANG3:10;
    then
A5: still_not-bound_in r c= still_not-bound_in q '&' r by XBOOLE_1:7;
    still_not-bound_in q c= still_not-bound_in q '&' r by A4,XBOOLE_1:7;
    hence P[q,t,K,f] & P[r,t+QuantNbr(q),K,f] by A3,A5,XBOOLE_1:1;
  end;
A6: now
    let q,x,t,K,f such that
    [All(x,q),t,K,f] in SepQuadruples p and
A7: P[All(x,q),t,K,f];
    still_not-bound_in All(x,q) = still_not-bound_in q \ {x} by QC_LANG3:12;
    then still_not-bound_in q c= still_not-bound_in p \/ K \/ {x} by A7,
XBOOLE_1:44;
    hence P[q,t++,K \/ {x},f+*(x .--> x.t)] by XBOOLE_1:4;
  end;
A8: 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(A8,A1,A2,A6);
end;
