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 [#] 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
A1: f is_differentiable_on [#] REAL and
A2: for x0 being Real holds 0 <= diff (f,x0) ; :: thesis: f | ([#] REAL) is non-decreasing

now :: thesis: for r1, r2 being Real st r1 in ([#] REAL) /\ (dom f) & r2 in ([#] REAL) /\ (dom f) & r1 < r2 holds
f . r1 <= f . r2

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
A3: r1 in ([#] REAL) /\ (dom f) and
A4: r2 in ([#] REAL) /\ (dom f) and
A5: r1 < r2 ; :: thesis: f . r1 <= f . r2
set rx = max (r1,r2);
set rn = min (r1,r2);
A6: r2 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r2 - 0 by XREAL_1:15, XXREAL_0:17;
then r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < (max (r1,r2)) + 1 ) } by A6;
then A7: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def 2;
r2 in dom f by A4, XBOOLE_0:def 4;
then A8: r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A7, XBOOLE_0:def 4;
A9: for g1 being Real st g1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ holds
0 <= diff (f,g1) by A2;
f is_differentiable_on ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by A1, FDIFF_1:26;
then A10: f | ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ is non-decreasing by A9, ROLLE:11;
A11: r1 + 0 < (max (r1,r2)) + 1 by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1 < r1 - 0 by XREAL_1:15, XXREAL_0:17;
then r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < (max (r1,r2)) + 1 ) } by A11;
then A12: r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ by RCOMP_1:def 2;
r1 in dom f by A3, XBOOLE_0:def 4;
then r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f) by A12, XBOOLE_0:def 4;
hence f . r1 <= f . r2 by A5, A10, A8, RFUNCT_2:22; :: thesis: verum

end;

hence f | ([#] REAL) is non-decreasing by RFUNCT_2:22; :: thesis: verum