let p be Real; :: thesis: for f being one-to-one PartFunc of REAL,REAL st ( f | is increasing or f | is decreasing ) & right_open_halfline p c= dom f holds
((f | ) ") | () is continuous

let f be one-to-one PartFunc of REAL,REAL; :: thesis: ( ( f | is increasing or f | is decreasing ) & right_open_halfline p c= dom f implies ((f | ) ") | () is continuous )
set L = right_open_halfline p;
assume that
A1: ( f | is increasing or f | is decreasing ) and
A2: right_open_halfline p c= dom f ; :: thesis: ((f | ) ") | () is continuous
now :: thesis: ((f | ) ") | () is continuous
per cases ( f | is increasing or f | is decreasing ) by A1;
suppose A3: f | is increasing ; :: thesis: ((f | ) ") | () is continuous
A4: now :: thesis: for r being Element of REAL st r in f .: holds
r in dom ((f | ) ")
let r be Element of REAL ; :: thesis: ( r in f .: implies r in dom ((f | ) ") )
assume r in f .: ; :: thesis: r in dom ((f | ) ")
then consider s being Element of REAL such that
A5: ( s in dom f & s in right_open_halfline p ) and
A6: r = f . s by PARTFUN2:59;
s in (dom f) /\ by ;
then A7: s in dom (f | ) by RELAT_1:61;
then r = (f | ) . s by ;
then r in rng (f | ) by ;
hence r in dom ((f | ) ") by FUNCT_1:33; :: thesis: verum
end;
A8: ((f | ) ") .: () = ((f | ) ") .: (rng (f | )) by RELAT_1:115
.= ((f | ) ") .: (dom ((f | ) ")) by FUNCT_1:33
.= rng ((f | ) ") by RELAT_1:113
.= dom (f | ) by FUNCT_1:33
.= (dom f) /\ by RELAT_1:61
.= right_open_halfline p by ;
((f | ) ") | () is increasing by ;
hence ((f | ) ") | () is continuous by ; :: thesis: verum
end;
suppose A9: f | is decreasing ; :: thesis: ((f | ) ") | () is continuous
A10: now :: thesis: for r being Element of REAL st r in f .: holds
r in dom ((f | ) ")
let r be Element of REAL ; :: thesis: ( r in f .: implies r in dom ((f | ) ") )
assume r in f .: ; :: thesis: r in dom ((f | ) ")
then consider s being Element of REAL such that
A11: ( s in dom f & s in right_open_halfline p ) and
A12: r = f . s by PARTFUN2:59;
s in (dom f) /\ by ;
then A13: s in dom (f | ) by RELAT_1:61;
then r = (f | ) . s by ;
then r in rng (f | ) by ;
hence r in dom ((f | ) ") by FUNCT_1:33; :: thesis: verum
end;
A14: ((f | ) ") .: () = ((f | ) ") .: (rng (f | )) by RELAT_1:115
.= ((f | ) ") .: (dom ((f | ) ")) by FUNCT_1:33
.= rng ((f | ) ") by RELAT_1:113
.= dom (f | ) by FUNCT_1:33
.= (dom f) /\ by RELAT_1:61
.= right_open_halfline p by ;
((f | ) ") | () is decreasing by ;
hence ((f | ) ") | () is continuous by ; :: thesis: verum
end;
end;
end;
hence ((f | ) ") | () is continuous ; :: thesis: verum