for r being Real
for f being PartFunc of REAL,REAL st right_open_halfline r c= dom f & f is_differentiable_on right_open_halfline r & ( for x0 being Real st x0 in right_open_halfline r holds
0 < diff (f,x0) ) holds
( f | (right_open_halfline r) is increasing & f | (right_open_halfline r) is one-to-one )