let f be PartFunc of REAL ,REAL ; :: thesis: ( [#] REAL c= dom f & f is_differentiable_on [#] REAL & ( for x0 being Real holds 0 <= diff f,x0 ) implies f | ([#] REAL ) is non-decreasing )
assume A1: [#] REAL c= dom f ; :: thesis: ( not f is_differentiable_on [#] REAL or ex x0 being Real st not 0 <= diff f,x0 or f | ([#] REAL ) is non-decreasing )
assume that
A2: f is_differentiable_on [#] REAL and
A3: for x0 being Real holds 0 <= diff f,x0 ; :: thesis: f | ([#] REAL ) is non-decreasing

now

let r1, r2 be Real; :: thesis: ( r1 in ([#] REAL ) /\ (dom f) & r2 in ([#] REAL ) /\ (dom f) & r1 < r2 implies f . r1 <= f . r2 )
assume that
A4: r1 in ([#] REAL ) /\ (dom f) and
A5: r2 in ([#] REAL ) /\ (dom f) and
A6: r1 < r2 ; :: thesis: f . r1 <= f . r2
set rx = max r1,r2;
set rn = min r1,r2;
A7: r2 + 0 < (max r1,r2) + 1 by XREAL_1:10, XXREAL_0:25;
(min r1,r2) - 1 < r2 - 0 by XREAL_1:17, XXREAL_0:17;
then r2 in { g2 where g2 is Real : ( (min r1,r2) - 1 < g2 & g2 < (max r1,r2) + 1 ) } by A7;
then A8: r2 in ].((min r1,r2) - 1),((max r1,r2) + 1).[ by RCOMP_1:def 2;
r2 in dom f by A5, XBOOLE_0:def 4;
then A9: r2 in ].((min r1,r2) - 1),((max r1,r2) + 1).[ /\ (dom f) by A8, XBOOLE_0:def 4;
A10: for g1 being Real st g1 in ].((min r1,r2) - 1),((max r1,r2) + 1).[ holds
0 <= diff f,g1 by A3;
f is_differentiable_on ].((min r1,r2) - 1),((max r1,r2) + 1).[ by A2, FDIFF_1:34;
then A11: f | ].((min r1,r2) - 1),((max r1,r2) + 1).[ is non-decreasing by A1, A10, ROLLE:11, XBOOLE_1:1;
A12: r1 + 0 < (max r1,r2) + 1 by XREAL_1:10, XXREAL_0:25;
(min r1,r2) - 1 < r1 - 0 by XREAL_1:17, XXREAL_0:17;
then r1 in { g1 where g1 is Real : ( (min r1,r2) - 1 < g1 & g1 < (max r1,r2) + 1 ) } by A12;
then A13: r1 in ].((min r1,r2) - 1),((max r1,r2) + 1).[ by RCOMP_1:def 2;
r1 in dom f by A4, XBOOLE_0:def 4;
then r1 in ].((min r1,r2) - 1),((max r1,r2) + 1).[ /\ (dom f) by A13, XBOOLE_0:def 4;
hence f . r1 <= f . r2 by A6, A11, A9, RFUNCT_2:45; :: thesis: verum

end;

hence f | ([#] REAL ) is non-decreasing by RFUNCT_2:45; :: thesis: verum