set L = [#] REAL ;
let f be one-to-one PartFunc of REAL ,REAL ; :: thesis: ( ( f | ([#] REAL ) is increasing or f | ([#] REAL ) is decreasing ) & f is total implies (f " ) | (rng f) is continuous )
assume that
A1: ( f | ([#] REAL ) is increasing or f | ([#] REAL ) is decreasing ) and
A2: f is total ; :: thesis: (f " ) | (rng f) is continuous
A3: dom f = [#] REAL by A2, PARTFUN1:def 4;

now

per cases ( f | ([#] REAL ) is increasing or f | ([#] REAL ) is decreasing ) by A1;

suppose A4: f | ([#] REAL ) is increasing ; :: thesis: (f " ) | (rng f) is continuous

A5: now

let r be Real; :: thesis: ( r in f .: ([#] REAL ) implies r in dom ((f | ([#] REAL )) " ) )
assume r in f .: ([#] REAL ) ; :: thesis: r in dom ((f | ([#] REAL )) " )
then consider s being Real such that
A6: s in dom f and
s in [#] REAL and
A7: r = f . s by PARTFUN2:78;
s in (dom f) /\ ([#] REAL ) by A6, XBOOLE_0:def 4;
then A8: s in dom (f | ([#] REAL )) by RELAT_1:90;
then r = (f | ([#] REAL )) . s by A7, FUNCT_1:70;
then r in rng (f | ([#] REAL )) by A8, FUNCT_1:def 5;
hence r in dom ((f | ([#] REAL )) " ) by FUNCT_1:55; :: thesis: verum

end;

A9: ((f | ([#] REAL )) " ) .: (f .: ([#] REAL )) = ((f | ([#] REAL )) " ) .: (rng (f | ([#] REAL ))) by RELAT_1:148
.= ((f | ([#] REAL )) " ) .: (dom ((f | ([#] REAL )) " )) by FUNCT_1:55
.= rng ((f | ([#] REAL )) " ) by RELAT_1:146
.= dom (f | ([#] REAL )) by FUNCT_1:55
.= (dom f) /\ ([#] REAL ) by RELAT_1:90
.= REAL by A3 ;
((f | ([#] REAL )) " ) | (f .: ([#] REAL )) is increasing by A4, Th17;
then ((f | ([#] REAL )) " ) | (f .: ([#] REAL )) is continuous by A5, A9, Th24, SUBSET_1:7;
then ((f | ([#] REAL )) " ) | (rng f) is continuous by A3, RELAT_1:146;
hence (f " ) | (rng f) is continuous by A3, RELAT_1:97; :: thesis: verum

end;

suppose A10: f | ([#] REAL ) is decreasing ; :: thesis: (f " ) | (rng f) is continuous

A11: now

let r be Real; :: thesis: ( r in f .: ([#] REAL ) implies r in dom ((f | ([#] REAL )) " ) )
assume r in f .: ([#] REAL ) ; :: thesis: r in dom ((f | ([#] REAL )) " )
then consider s being Real such that
A12: s in dom f and
s in [#] REAL and
A13: r = f . s by PARTFUN2:78;
s in (dom f) /\ ([#] REAL ) by A12, XBOOLE_0:def 4;
then A14: s in dom (f | ([#] REAL )) by RELAT_1:90;
then r = (f | ([#] REAL )) . s by A13, FUNCT_1:70;
then r in rng (f | ([#] REAL )) by A14, FUNCT_1:def 5;
hence r in dom ((f | ([#] REAL )) " ) by FUNCT_1:55; :: thesis: verum

end;

A15: ((f | ([#] REAL )) " ) .: (f .: ([#] REAL )) = ((f | ([#] REAL )) " ) .: (rng (f | ([#] REAL ))) by RELAT_1:148
.= ((f | ([#] REAL )) " ) .: (dom ((f | ([#] REAL )) " )) by FUNCT_1:55
.= rng ((f | ([#] REAL )) " ) by RELAT_1:146
.= dom (f | ([#] REAL )) by FUNCT_1:55
.= (dom f) /\ ([#] REAL ) by RELAT_1:90
.= REAL by A3 ;
((f | ([#] REAL )) " ) | (f .: ([#] REAL )) is decreasing by A10, Th18;
then ((f | ([#] REAL )) " ) | (f .: ([#] REAL )) is continuous by A11, A15, Th24, SUBSET_1:7;
then ((f | ([#] REAL )) " ) | (rng f) is continuous by A3, RELAT_1:146;
hence (f " ) | (rng f) is continuous by A3, RELAT_1:97; :: thesis: verum

end;

end;

end;

hence (f " ) | (rng f) is continuous ; :: thesis: verum