
theorem
for f be PartFunc of REAL,REAL, I be non empty Interval st I is open_interval
 holds f is_differentiable_on I iff f is_differentiable_on_interval I
proof
    let f be PartFunc of REAL,REAL, I be non empty Interval;
    assume I is open_interval; then
    consider a,b be R_eal such that
A1:  I = ].a,b.[ by MEASURE5:def 2;

A2: a = inf I & b = sup I by A1,XXREAL_1:28,XXREAL_2:28,32;

    hence f is_differentiable_on I implies
     f is_differentiable_on_interval I by A1,XXREAL_1:4,28;
    assume f is_differentiable_on_interval I;
    hence f is_differentiable_on I by A1,A2;
end;
