let p, g be Real; :: thesis: for f being 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
rng (f | ].p,g.[) is open
let f be PartFunc of REAL ,REAL ; :: thesis: ( ].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 ) implies rng (f | ].p,g.[) is open )
assume A1: ].p,g.[ c= dom f ; :: thesis: ( not f is_differentiable_on ].p,g.[ or ( ex x0 being Real st
( x0 in ].p,g.[ & not 0 < diff f,x0 ) & ex x0 being Real st
( x0 in ].p,g.[ & not diff f,x0 < 0 ) ) or rng (f | ].p,g.[) is open )
assume that
A2: f is_differentiable_on ].p,g.[ and
A3: ( 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 ) ; :: thesis: rng (f | ].p,g.[) is open
A4: f | ].p,g.[ is continuous by A2, FDIFF_1:33;

now

per cases ( 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 ) by A3;

suppose A5: for x0 being Real st x0 in ].p,g.[ holds
0 < diff f,x0 ; :: thesis: rng (f | ].p,g.[) is open

then f | ].p,g.[ is increasing by A1, A2, ROLLE:9;
hence rng (f | ].p,g.[) is open by A1, A4, FCONT_3:31; :: thesis: verum

end;

suppose A6: for x0 being Real st x0 in ].p,g.[ holds
diff f,x0 < 0 ; :: thesis: rng (f | ].p,g.[) is open

then f | ].p,g.[ is decreasing by A1, A2, ROLLE:10;
hence rng (f | ].p,g.[) is open by A1, A4, FCONT_3:31; :: thesis: verum

end;

end;

end;

hence rng (f | ].p,g.[) is open ; :: thesis: verum