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

now

let r1, r2 be Real; :: thesis: ( r1 in (left_open_halfline r) /\ (dom f) & r2 in (left_open_halfline r) /\ (dom f) & r1 < r2 implies f . r2 < f . r1 )
assume that
A4: r1 in (left_open_halfline r) /\ (dom f) and
A5: r2 in (left_open_halfline r) /\ (dom f) and
A6: r1 < r2 ; :: thesis: f . r2 < f . r1
set rr = min r1,r2;
A7: (min r1,r2) - 1 < r2 - 0 by XREAL_1:17, XXREAL_0:17;
r2 in left_open_halfline r by A5, XBOOLE_0:def 4;
then r2 in { p where p is Real : p < r } by XXREAL_1:229;
then ex g2 being Real st
( g2 = r2 & g2 < r ) ;
then r2 in { g2 where g2 is Real : ( (min r1,r2) - 1 < g2 & g2 < r ) } by A7;
then A8: r2 in ].((min r1,r2) - 1),r.[ by RCOMP_1:def 2;
r2 in dom f by A5, XBOOLE_0:def 4;
then A9: r2 in ].((min r1,r2) - 1),r.[ /\ (dom f) by A8, XBOOLE_0:def 4;
A10: f is_differentiable_on ].((min r1,r2) - 1),r.[ by A2, FDIFF_1:34, XXREAL_1:263;
A11: ].((min r1,r2) - 1),r.[ c= left_open_halfline r by XXREAL_1:263;
then for g1 being Real st g1 in ].((min r1,r2) - 1),r.[ holds
diff f,g1 < 0 by A3;
then A12: f | ].((min r1,r2) - 1),r.[ is decreasing by A1, A11, A10, ROLLE:10, XBOOLE_1:1;
A13: (min r1,r2) - 1 < r1 - 0 by XREAL_1:17, XXREAL_0:17;
r1 in left_open_halfline r by A4, XBOOLE_0:def 4;
then r1 in { g where g is Real : g < r } by XXREAL_1:229;
then ex g1 being Real st
( g1 = r1 & g1 < r ) ;
then r1 in { g1 where g1 is Real : ( (min r1,r2) - 1 < g1 & g1 < r ) } by A13;
then A14: r1 in ].((min r1,r2) - 1),r.[ by RCOMP_1:def 2;
r1 in dom f by A4, XBOOLE_0:def 4;
then r1 in ].((min r1,r2) - 1),r.[ /\ (dom f) by A14, XBOOLE_0:def 4;
hence f . r2 < f . r1 by A6, A12, A9, RFUNCT_2:44; :: thesis: verum

end;

hence f | (left_open_halfline r) is decreasing by RFUNCT_2:44; :: thesis: f | (left_open_halfline r) is one-to-one
hence f | (left_open_halfline r) is one-to-one by FCONT_3:16; :: thesis: verum