
theorem Th11:
  for a be Real, A be Subset of REAL,
    rho be real-valued Function st A c= [.a,a.]
  holds vol(A,rho) = 0
  proof
    let a be Real, A be Subset of REAL,
        rho be real-valued Function;
    assume A c= [.a,a.]; then
    A c= {a} by XXREAL_1:17; then
    per cases by ZFMISC_1:33;
    suppose A = {};
      hence vol(A,rho) = 0 by INTEGR22:def 1;
    end;
    suppose A = {a};
      hence vol(A,rho)
              = rho.(upper_bound {a}) - rho.(lower_bound {a}) by INTEGR22:def 1
             .= rho.(upper_bound {a}) - rho.(upper_bound {a}) by SEQ_4:10
             .= 0;
    end;
  end;
