let p be Real; :: thesis: for f being one-to-one PartFunc of REAL,REAL st ( f | (left_open_halfline p) is increasing or f | (left_open_halfline p) is decreasing ) & left_open_halfline p c= dom f holds
((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous
let f be one-to-one PartFunc of REAL,REAL; :: thesis: ( ( f | (left_open_halfline p) is increasing or f | (left_open_halfline p) is decreasing ) & left_open_halfline p c= dom f implies ((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous )
set L = left_open_halfline p;
assume that
A1: ( f | (left_open_halfline p) is increasing or f | (left_open_halfline p) is decreasing ) and
A2: left_open_halfline p c= dom f ; :: thesis: ((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous

now :: thesis: ((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous

per cases ( f | (left_open_halfline p) is increasing or f | (left_open_halfline p) is decreasing ) by A1;

suppose A3: f | (left_open_halfline p) is increasing ; :: thesis: ((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous

A4: now :: thesis: for r being Element of REAL st r in f .: (left_open_halfline p) holds
r in dom ((f | (left_open_halfline p)) ")

let r be Element of REAL ; :: thesis: ( r in f .: (left_open_halfline p) implies r in dom ((f | (left_open_halfline p)) ") )
assume r in f .: (left_open_halfline p) ; :: thesis: r in dom ((f | (left_open_halfline p)) ")
then consider s being Element of REAL such that
A5: ( s in dom f & s in left_open_halfline p ) and
A6: r = f . s by PARTFUN2:59;
s in (dom f) /\ (left_open_halfline p) by A5, XBOOLE_0:def 4;
then A7: s in dom (f | (left_open_halfline p)) by RELAT_1:61;
then r = (f | (left_open_halfline p)) . s by A6, FUNCT_1:47;
then r in rng (f | (left_open_halfline p)) by A7, FUNCT_1:def 3;
hence r in dom ((f | (left_open_halfline p)) ") by FUNCT_1:33; :: thesis: verum

end;

A8: ((f | (left_open_halfline p)) ") .: (f .: (left_open_halfline p)) = ((f | (left_open_halfline p)) ") .: (rng (f | (left_open_halfline p))) by RELAT_1:115
.= ((f | (left_open_halfline p)) ") .: (dom ((f | (left_open_halfline p)) ")) by FUNCT_1:33
.= rng ((f | (left_open_halfline p)) ") by RELAT_1:113
.= dom (f | (left_open_halfline p)) by FUNCT_1:33
.= (dom f) /\ (left_open_halfline p) by RELAT_1:61
.= left_open_halfline p by A2, XBOOLE_1:28 ;
((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is increasing by A3, Th9;
hence ((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous by A4, A8, Th11, SUBSET_1:2; :: thesis: verum

end;

suppose A9: f | (left_open_halfline p) is decreasing ; :: thesis: ((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous

A10: now :: thesis: for r being Element of REAL st r in f .: (left_open_halfline p) holds
r in dom ((f | (left_open_halfline p)) ")

let r be Element of REAL ; :: thesis: ( r in f .: (left_open_halfline p) implies r in dom ((f | (left_open_halfline p)) ") )
assume r in f .: (left_open_halfline p) ; :: thesis: r in dom ((f | (left_open_halfline p)) ")
then consider s being Element of REAL such that
A11: ( s in dom f & s in left_open_halfline p ) and
A12: r = f . s by PARTFUN2:59;
s in (dom f) /\ (left_open_halfline p) by A11, XBOOLE_0:def 4;
then A13: s in dom (f | (left_open_halfline p)) by RELAT_1:61;
then r = (f | (left_open_halfline p)) . s by A12, FUNCT_1:47;
then r in rng (f | (left_open_halfline p)) by A13, FUNCT_1:def 3;
hence r in dom ((f | (left_open_halfline p)) ") by FUNCT_1:33; :: thesis: verum

end;

A14: ((f | (left_open_halfline p)) ") .: (f .: (left_open_halfline p)) = ((f | (left_open_halfline p)) ") .: (rng (f | (left_open_halfline p))) by RELAT_1:115
.= ((f | (left_open_halfline p)) ") .: (dom ((f | (left_open_halfline p)) ")) by FUNCT_1:33
.= rng ((f | (left_open_halfline p)) ") by RELAT_1:113
.= dom (f | (left_open_halfline p)) by FUNCT_1:33
.= (dom f) /\ (left_open_halfline p) by RELAT_1:61
.= left_open_halfline p by A2, XBOOLE_1:28 ;
((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is decreasing by A9, Th10;
hence ((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous by A10, A14, Th11, SUBSET_1:2; :: thesis: verum

end;

end;

end;

hence ((f | (left_open_halfline p)) ") | (f .: (left_open_halfline p)) is continuous ; :: thesis: verum