
theorem
for f be PartFunc of REAL,REAL, a,b be Real st a < b &
 f is_differentiable_on_interval ['a,b'] holds
 f is_differentiable_on ].a,b.[
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume that
A1:  a < b and
A2:  f is_differentiable_on_interval ['a,b'];

A3: [.a,b.] = ['a,b'] by A1,INTEGRA5:def 3;

    a = inf [.a,b.] & b = sup [.a,b.] by A1,XXREAL_2:25,29;
    hence f is_differentiable_on ].a,b.[ by A2,A3;
end;
