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

now

per cases ( for x0 being Real st x0 in left_open_halfline p holds
0 < diff f,x0 or for x0 being Real st x0 in left_open_halfline p holds
diff f,x0 < 0 ) by A3;

suppose for x0 being Real st x0 in left_open_halfline p holds
0 < diff f,x0 ; :: thesis: rng (f | (left_open_halfline p)) is open

then f | (left_open_halfline p) is increasing by A1, A2, Th29;
hence rng (f | (left_open_halfline p)) is open by A1, A4, FCONT_3:32; :: thesis: verum

end;

suppose for x0 being Real st x0 in left_open_halfline p holds
diff f,x0 < 0 ; :: thesis: rng (f | (left_open_halfline p)) is open

then f | (left_open_halfline p) is decreasing by A1, A2, Th30;
hence rng (f | (left_open_halfline p)) is open by A1, A4, FCONT_3:32; :: thesis: verum

end;

end;

end;

hence rng (f | (left_open_halfline p)) is open ; :: thesis: verum