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 Th39:
  [q,t,K,f] in SepQuadruples p & x.u in f.:K implies u < t
proof
  defpred P[Element of CQC-WFF(A),QC-symbol of A, Element of Fin
  bound_QC-variables(A),Function] means for u holds x.u in $4.:$3 implies
   u < $2;
A1: for q,v,K,f st ['not' q,v,K,f] in SepQuadruples p & P['not' q,v,K,f]
  holds P[q,v,K,f];
A2: now
    let q,r,v,K,f;
    assume [q '&' r, v, K,f] in SepQuadruples p;
    assume
A3: P[q '&' r, v, K,f];
    hence P[q,v,K,f];
    thus P[r,v+QuantNbr(q),K,f]
    proof
      let u;
A4:   v <= v + QuantNbr(q) by QC_LANG1:31;
      assume
       x.u in f.:K;
      hence thesis by A3,A4,QC_LANG1:30;
    end;
  end;
A5: now
    let q,x,v,K,f such that
    [All(x,q),v,K,f] in SepQuadruples p;
    assume
A6: P[All(x,q),v,K,f];
    thus P[q,v++,K \/ {.x .},f+*(x .--> x.v)]
    proof
      let u;
      assume x.u in (f+*(x .--> x.v)).:(K \/ {x});
      then x.u in (f+*(x .--> x.v)).:K \/ (f+*(x .--> x.v)).: {x}
       by RELAT_1:120;
      then
A7:   x.u in (f+*(x .--> x.v)).:K or x.u
       in Im(f+*(x .--> x.v),x) by XBOOLE_0:def 3;
      (f+*(x .--> x.v)).:K c= f.:K \/ {x.v} by Th2;
      then x.u in f.:K or x.u in {x.v} by A7,Th1,XBOOLE_0:def 3;
      then u < v or x.u = x.v by A6,TARSKI:def 1;
      then u < v or u = v by XTUPLE_0:1;
      then u <= v & v < v++ by QC_LANG1:22,27,def 34;
      hence thesis by QC_LANG1:29;
    end;
  end;
A8: P[p, index p,{}.bound_QC-variables(A),id bound_QC-variables(A)];
  for q,v,K,f st [q,v,K,f] in SepQuadruples p holds P[q,v,K,f] from
  Sepregression(A8,A1,A2,A5);
  hence thesis;
end;
