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

now

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

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

then ((f | ].p,g.[) " ) | (f .: ].p,g.[) is increasing by Th17;
then ((f | ].p,g.[) " ) | (f .: ].p,g.[) is non-decreasing ;
then A3: ((f | ].p,g.[) " ) | (f .: ].p,g.[) is monotone ;

now

let r be Real; :: thesis: ( r in f .: ].p,g.[ implies r in dom ((f | ].p,g.[) " ) )
assume r in f .: ].p,g.[ ; :: thesis: r in dom ((f | ].p,g.[) " )
then consider s being Real such that
A4: ( s in dom f & s in ].p,g.[ & r = f . s ) by PARTFUN2:78;
( s in (dom f) /\ ].p,g.[ & r = f . s ) by A4, XBOOLE_0:def 4;
then ( s in dom (f | ].p,g.[) & r = f . s ) by RELAT_1:90;
then ( s in dom (f | ].p,g.[) & r = (f | ].p,g.[) . s ) by FUNCT_1:70;
then r in rng (f | ].p,g.[) by FUNCT_1:def 5;
hence r in dom ((f | ].p,g.[) " ) by FUNCT_1:55; :: thesis: verum

end;

then A5: f .: ].p,g.[ c= dom ((f | ].p,g.[) " ) by SUBSET_1:7;
((f | ].p,g.[) " ) .: (f .: ].p,g.[) = ((f | ].p,g.[) " ) .: (rng (f | ].p,g.[)) by RELAT_1:148
.= ((f | ].p,g.[) " ) .: (dom ((f | ].p,g.[) " )) by FUNCT_1:55
.= rng ((f | ].p,g.[) " ) by RELAT_1:146
.= dom (f | ].p,g.[) by FUNCT_1:55
.= (dom f) /\ ].p,g.[ by RELAT_1:90
.= ].p,g.[ by A2, XBOOLE_1:28 ;
hence ((f | ].p,g.[) " ) | (f .: ].p,g.[) is continuous by A3, A5, Th23; :: thesis: verum

end;

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

then ((f | ].p,g.[) " ) | (f .: ].p,g.[) is decreasing by Th18;
then ((f | ].p,g.[) " ) | (f .: ].p,g.[) is non-increasing ;
then A6: ((f | ].p,g.[) " ) | (f .: ].p,g.[) is monotone ;

now

let r be Real; :: thesis: ( r in f .: ].p,g.[ implies r in dom ((f | ].p,g.[) " ) )
assume r in f .: ].p,g.[ ; :: thesis: r in dom ((f | ].p,g.[) " )
then consider s being Real such that
A7: ( s in dom f & s in ].p,g.[ & r = f . s ) by PARTFUN2:78;
s in (dom f) /\ ].p,g.[ by A7, XBOOLE_0:def 4;
then s in dom (f | ].p,g.[) by RELAT_1:90;
then ( s in dom (f | ].p,g.[) & r = (f | ].p,g.[) . s ) by A7, FUNCT_1:70;
then r in rng (f | ].p,g.[) by FUNCT_1:def 5;
hence r in dom ((f | ].p,g.[) " ) by FUNCT_1:55; :: thesis: verum

end;

then A8: f .: ].p,g.[ c= dom ((f | ].p,g.[) " ) by SUBSET_1:7;
((f | ].p,g.[) " ) .: (f .: ].p,g.[) = ((f | ].p,g.[) " ) .: (rng (f | ].p,g.[)) by RELAT_1:148
.= ((f | ].p,g.[) " ) .: (dom ((f | ].p,g.[) " )) by FUNCT_1:55
.= rng ((f | ].p,g.[) " ) by RELAT_1:146
.= dom (f | ].p,g.[) by FUNCT_1:55
.= (dom f) /\ ].p,g.[ by RELAT_1:90
.= ].p,g.[ by A2, XBOOLE_1:28 ;
hence ((f | ].p,g.[) " ) | (f .: ].p,g.[) is continuous by A6, A8, Th23; :: thesis: verum

end;

end;

end;

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