
theorem
  for A being Interval, x,y being Real st 0 <= x & y
  = diameter(A) holds x * y = diameter(x ** A)
proof
  let A be Interval;
  let x,y be Real such that
A1: 0 <= x and
A2: y = diameter(A);
  per cases;
  suppose
A3: A is empty;
    then
A4: x ** A is empty;
    thus x * y = 0 by A2,A3,MEASURE5:10
      .= diameter(x ** A) by A4,MEASURE5:10;
  end;
  suppose
A5: A is non empty;
    then consider z being Real such that
A6: z in A;
    reconsider z as Real;
A7: x * z in x ** A by A6,MEMBER_1:193;
    reconsider AA = A as non empty Subset of REAL by A5;
    reconsider u = x as R_eal by XXREAL_0:def 1;
A8: inf(x ** AA) = u * inf AA by A1,Th19;
    reconsider z = x as R_eal by XXREAL_0:def 1;
    thus x * y = z * diameter A by A2,EXTREAL1:1
      .= z * (sup A - inf A) by A5,MEASURE5:def 6
      .= z * sup A - (z * inf A) by XXREAL_3:100
      .= sup(x ** A) - inf(x ** A) by A1,A8,Th18
      .= diameter(x ** A) by A7,MEASURE5:def 6;
  end;
end;
