reserve x for object,
  a,b for Real,
  k,k1,i1,j1,w for Nat,
  m,m1,n,n1 for Integer;
reserve p,q for Rational;

theorem Th40:
  denominator(-p) = denominator(p) & numerator(-p) = -numerator(p)
proof
A2: p=-(-p) .=-numerator(-p)/denominator(-p) by Th12
    .=(-numerator(-p))/denominator(-p);
  consider w such that
  -numerator(-p)=numerator(p)*w and
A3: denominator(-p)=denominator(p)*w by A2,Th24;
  -p=-numerator(p)/denominator(p) by Th12
    .=(-numerator(p))/denominator(p);
  then consider k such that
A4: -numerator(p)=numerator(-p)*k and
A5: denominator(p)=denominator(-p)*k by Th24;
  denominator p = denominator(p)*w*k by A5,A3
    .= denominator(p)*(w*k);
  then
A6: k = 1 by NAT_1:15,XCMPLX_1:7;
  hence denominator(p)=denominator(-p) by A5;
  thus thesis by A4,A6;
end;
