reserve a,b for R_eal;
reserve A,B for Interval;

theorem
  A c= B & B =[.a,b.] & b <= a implies diameter(A) = 0. & diameter(B) = 0.
proof
  assume that
A1: A c= B and
A2: B =[.a,b.] and
A3: b <= a;
  per cases by A3,XXREAL_0:1;
  suppose
A4: a = b;
    then B = {a} by A2,XXREAL_1:17;
    then inf B = a & sup B = a by XXREAL_2:11,13;
    then
A5: diameter B = a - a by A2,A4,Def6
      .= 0 by XXREAL_3:7;
    then diameter A <= 0 by A1,Lm2;
    hence thesis by A5,Lm1;
  end;
  suppose
    b < a;
    then
A6: B = {} by A2,XXREAL_1:29;
    then A = {} by A1;
    hence thesis by A6,Def6;
  end;
end;
