reserve Al for QC-alphabet;
reserve i,j,k for Nat,
  A,D for non empty set;
reserve f1,f2 for Element of Funcs(Valuations_in(Al,A),BOOLEAN),
  x,x1,y for bound_QC-variable of Al,
  v,v1 for Element of Valuations_in(Al,A);
reserve ll for CQC-variable_list of k,Al;
reserve p,q,s,t for Element of CQC-WFF(Al),
  J for interpretation of Al,A,
  P for QC-pred_symbol of k,Al,
  r for Element of relations_on A;
reserve u,w,z for Element of BOOLEAN;
reserve w,v2 for Element of Valuations_in(Al,A),
  z for bound_QC-variable of Al;
reserve u,w for Element of Valuations_in(Al,A);
reserve s9 for QC-formula of Al;

theorem
  for s being QC-formula of Al st p = s.x & q = s.y & not x in
  still_not-bound_in s & J |= p holds J |= q
proof
  let s be QC-formula of Al;
  assume that
A1: p = s.x and
A2: q = s.y and
A3: not x in still_not-bound_in s and
A4: J |= p;
  now
    assume
A5: x <> y;
A6: now
      let u;
      consider w being Element of Valuations_in(Al,A) such that
A7:   (for z being bound_QC-variable of Al st z <> x holds w.z = u.z) & w.x
      = u.y by Lm3;
      w.x = w.y by A7;
      then
A8:   Valid(p,J).w = Valid(q,J).w by A1,A2,Th30;
      J,w |= p by A4;
      then
A9:   Valid(p,J).w = TRUE;
      not x in still_not-bound_in q by A2,A3,A5,Th31;
      hence Valid(q,J).u = TRUE by A7,A8,A9,Th27;
    end;
    for v holds J,v |= q by A6;
    hence thesis;
  end;
  hence thesis by A1,A2,A4;
end;
