
theorem
  for A being Interval, x being Real holds
    diameter(A) = diameter(x ++ A)
proof
  let A be Interval, x be Real;
  per cases;
  suppose A is empty;
    hence thesis;
  end;
  suppose
A1: A is non empty;
    then consider y being Real such that
A2: y in A;
    reconsider y as Real;
A3: x + y in x ++ A by A2,Lm1;
    reconsider y = x as R_eal by XXREAL_0:def 1;
    thus diameter(A) = sup A - inf A by A1,MEASURE5:def 6
      .= y + sup A - (y + inf A) by XXREAL_3:33
      .= sup(x ++ A) - (y + inf A) by Th31
      .= sup(x ++ A) - inf(x ++ A) by Th32
      .= diameter(x ++ A) by A3,MEASURE5:def 6;
  end;
end;
