reserve A for QC-alphabet;
reserve p, q, r, s, p1, q1 for Element of CQC-WFF(A),
  X, Y, Z, X1, X2 for Subset of CQC-WFF(A),
  h for QC-formula of A,
  x, y for bound_QC-variable of A,
  n for Element of NAT;

theorem Th7:
  X |- Y iff Y c= Cn(X)
proof
  hereby
    assume
A1: X |- Y;
    now
      let p be object;
      assume
A2:   p in Y;
      then reconsider p9 = p as Element of CQC-WFF(A);
      X |- p9 by A1,A2;
      hence p in Cn(X) by CQC_THE1:def 8;
    end;
    hence Y c= Cn(X);
  end;
  thus thesis by CQC_THE1:def 8;
end;
