let a, b be R_eal; :: thesis: ( ( a < b implies diameter [.a,b.[ = b - a ) & ( b <= a implies diameter [.a,b.[ = 0. ) )

hereby :: thesis: ( b <= a implies diameter [.a,b.[ = 0. )

assume Z: a < b ; :: thesis: diameter [.a,b.[ = b - a
then A: [.a,b.[ <> {} by XXREAL_1:31;
( inf [.a,b.[ = a & sup [.a,b.[ = b ) by Z, XXREAL_2:26, XXREAL_2:31;
hence diameter [.a,b.[ = b - a by A, Def10; :: thesis: verum

end;

assume b <= a ; :: thesis: diameter [.a,b.[ = 0.
then [.a,b.[ = {} by XXREAL_1:27;
hence diameter [.a,b.[ = 0. by Def10; :: thesis: verum