let g, p be Real; :: thesis: for f being one-to-one PartFunc of REAL,REAL st p <= g & [.p,g.] c= dom f & f | [.p,g.] is continuous holds
(f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous
let f be one-to-one PartFunc of REAL,REAL; :: thesis: ( p <= g & [.p,g.] c= dom f & f | [.p,g.] is continuous implies (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous )
assume that
A1: p <= g and
A2: [.p,g.] c= dom f and
A3: f | [.p,g.] is continuous ; :: thesis: (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous

now :: thesis: (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous

per cases ( f | [.p,g.] is increasing or f | [.p,g.] is decreasing ) by A1, A2, A3, Th17;

suppose f | [.p,g.] is increasing ; :: thesis: (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous

then ((f | [.p,g.]) ") | (f .: [.p,g.]) is increasing by RFUNCT_2:51;
then ((f ") | (f .: [.p,g.])) | (f .: [.p,g.]) is increasing by RFUNCT_2:17;
then (f ") | (f .: [.p,g.]) is monotone by RELAT_1:72;
then A4: (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is monotone by A1, A2, A3, Th19;
(f ") .: [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] = (f ") .: (f .: [.p,g.]) by A1, A2, A3, Th19
.= ((f ") | (f .: [.p,g.])) .: (f .: [.p,g.]) by RELAT_1:129
.= ((f | [.p,g.]) ") .: (f .: [.p,g.]) by RFUNCT_2:17
.= ((f | [.p,g.]) ") .: (rng (f | [.p,g.])) by RELAT_1:115
.= ((f | [.p,g.]) ") .: (dom ((f | [.p,g.]) ")) by FUNCT_1:33
.= rng ((f | [.p,g.]) ") by RELAT_1:113
.= dom (f | [.p,g.]) by FUNCT_1:33
.= (dom f) /\ [.p,g.] by RELAT_1:61
.= [.p,g.] by A2, XBOOLE_1:28 ;
hence (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous by A1, A4, FCONT_1:46; :: thesis: verum

end;

suppose f | [.p,g.] is decreasing ; :: thesis: (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous

then ((f | [.p,g.]) ") | (f .: [.p,g.]) is decreasing by RFUNCT_2:52;
then ((f ") | (f .: [.p,g.])) | (f .: [.p,g.]) is decreasing by RFUNCT_2:17;
then (f ") | (f .: [.p,g.]) is monotone by RELAT_1:72;
then A5: (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is monotone by A1, A2, A3, Th19;
(f ") .: [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] = (f ") .: (f .: [.p,g.]) by A1, A2, A3, Th19
.= ((f ") | (f .: [.p,g.])) .: (f .: [.p,g.]) by RELAT_1:129
.= ((f | [.p,g.]) ") .: (f .: [.p,g.]) by RFUNCT_2:17
.= ((f | [.p,g.]) ") .: (rng (f | [.p,g.])) by RELAT_1:115
.= ((f | [.p,g.]) ") .: (dom ((f | [.p,g.]) ")) by FUNCT_1:33
.= rng ((f | [.p,g.]) ") by RELAT_1:113
.= dom (f | [.p,g.]) by FUNCT_1:33
.= (dom f) /\ [.p,g.] by RELAT_1:61
.= [.p,g.] by A2, XBOOLE_1:28 ;
hence (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous by A1, A5, FCONT_1:46; :: thesis: verum

end;

end;

end;

hence (f ") | [.(lower_bound (f .: [.p,g.])),(upper_bound (f .: [.p,g.])).] is continuous ; :: thesis: verum