
theorem
for f be PartFunc of REAL,REAL, a,b be Real st a <= b &
 f is_differentiable_on_interval ['a,b'] holds
 (f`\['a,b']).a = Rdiff(f,a) & (f`\['a,b']).b = Ldiff(f,b) &
 for x be Real st x in ].a,b.[ holds (f`\['a,b']).x = diff(f,x)
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 = inf [.a,b.] & b = sup [.a,b.] by A1,XXREAL_2:25,29; then
    reconsider I=[.a,b.] as non empty left_end right_end real-membered set
       by A1,XXREAL_1:1,XXREAL_2:def 5,def 6;
A4: I = ['a,b'] by A1,INTEGRA5:def 3;

A5: a in I & b in I by A1,XXREAL_1:1;
    hence (f`\['a,b']).a = Rdiff(f,a) by A3,A2,A4,Def2;
    thus (f`\['a,b']).b = Ldiff(f,b) by A2,A5,A4,A3,Def2;
    thus for x be Real st x in ].a,b.[ holds (f`\['a,b']).x = diff(f,x)
    proof
     let x be Real;
     assume A6: x in ].a,b.[; then
A7:  x <> a & x <> b by XXREAL_1:4;
     ].a,b.[ c= [.a,b.] by XXREAL_1:37;
     hence thesis by A2,A3,A7,A4,A6,Def2;
    end;
end;
