theorem :: FDIFF_2:48

for g, p being Real
for f being one-to-one PartFunc of REAL,REAL st ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x0 being Real st x0 in ].p,g.[ holds
0 < diff (f,x0) or for x0 being Real st x0 in ].p,g.[ holds
diff (f,x0) < 0 ) holds
( f | ].p,g.[ is one-to-one & (f | ].p,g.[) " is_differentiable_on dom ((f | ].p,g.[) ") & ( for x0 being Real st x0 in dom ((f | ].p,g.[) ") holds
diff (((f | ].p,g.[) "),x0) = 1 / (diff (f,(((f | ].p,g.[) ") . x0))) ) )