reserve A for QC-alphabet;
reserve i,j,k for Nat;
reserve f for Substitution of A;

theorem Th7:
  for P being QC-pred_symbol of k,A for l being QC-variable_list of
  k,A holds P!l is Element of CQC-WFF(A) iff
  { l.i : 1 <= i & i <= len l & l.i in
  free_QC-variables(A) } = {} & { l.j : 1 <= j & j <= len l & l.j in
  fixed_QC-variables(A) } = {}
proof
  let P be QC-pred_symbol of k,A;
  let l be QC-variable_list of k, A;
A1: Free(P!l) = { l.j : 1 <= j & j <= len l & l.j in free_QC-variables(A) } by
QC_LANG3:54;
  Fixed(P!l) = { l.i : 1 <= i & i <= len l & l.i in fixed_QC-variables(A) }
  by QC_LANG3:64;
  hence thesis by A1,Th4;
end;
