
theorem Th5:
for a,b be Real st a <= b
  holds B-Meas.([.a,b.]) = b-a & B-Meas.([.a,b.[) = b-a
      & B-Meas.(].a,b.]) = b-a & B-Meas.(].a,b.[) = b-a
      & L-Meas.([.a,b.]) = b-a & L-Meas.([.a,b.[) = b-a
      & L-Meas.(].a,b.]) = b-a & L-Meas.(].a,b.[) = b-a
proof
    let a,b be Real;
    assume
A1:  a <= b;
    reconsider a1=a, b1=b as R_eal by XXREAL_0:def 1;
    B-Meas.([.a,b.]) = diameter [.a,b.] & L-Meas.([.a,b.]) = diameter [.a,b.]
     by MEASUR12:71,76; then
A2: B-Meas.([.a,b.]) = b1-a1 & L-Meas.([.a,b.]) = b1-a1 by A1,MEASURE5:6;

A3: B-Meas.([.a,b.[) = diameter [.a1,b1.[
  & B-Meas.(].a,b.]) = diameter ].a1,b1.]
  & B-Meas.(].a,b.[) = diameter ].a1,b1.[
  & L-Meas.([.a,b.[) = diameter [.a1,b1.[
  & L-Meas.(].a,b.]) = diameter ].a1,b1.]
  & L-Meas.(].a,b.[) = diameter ].a1,b1.[ by MEASUR12:71,76;

    now assume a <> b; then
     a1 < b1 by A1,XXREAL_0:1; then
     B-Meas.([.a,b.[) = b1-a1 & B-Meas.(].a,b.]) = b1-a1
   & B-Meas.(].a,b.[) = b1-a1
   & L-Meas.([.a,b.[) = b1-a1 & L-Meas.(].a,b.]) = b1-a1
   & L-Meas.(].a,b.[) = b1-a1 by A3,MEASURE5:5,7,8;
     hence B-Meas.([.a,b.[) = b-a & B-Meas.(].a,b.]) = b-a
      & B-Meas.(].a,b.[) = b-a
      & L-Meas.([.a,b.[) = b-a & L-Meas.(].a,b.]) = b-a
      & L-Meas.(].a,b.[) = b-a by Lm1;
    end;
    hence thesis by A2,A3,Lm1,MEASURE5:5,7,8;
end;
