let X be set ; :: thesis: for f being PartFunc of REAL,REAL st f | X is continuous & f " {0} = {} holds
(f ^) | X is continuous
let f be PartFunc of REAL,REAL; :: thesis: ( f | X is continuous & f " {0} = {} implies (f ^) | X is continuous )
assume that
A1: f | X is continuous and
A2: f " {0} = {} ; :: thesis: (f ^) | X is continuous
A3: dom (f ^) = (dom f) \ {} by A2, RFUNCT_1:def 2
.= dom f ;
let r be Real; :: according to FCONT_1:def 2 :: thesis: ( r in dom ((f ^) | X) implies (f ^) | X is_continuous_in r )
assume A4: r in dom ((f ^) | X) ; :: thesis: (f ^) | X is_continuous_in r
then A5: r in dom (f ^) by RELAT_1:57;
r in X by A4;
then A6: r in dom (f | X) by A3, A5, RELAT_1:57;
then A7: f | X is_continuous_in r by A1;

now :: thesis: for s1 being Real_Sequence st rng s1 c= dom ((f ^) | X) & s1 is convergent & lim s1 = r holds
( ((f ^) | X) /* s1 is convergent & lim (((f ^) | X) /* s1) = ((f ^) | X) . r )

A8: now :: thesis: not f . r = 0

assume f . r = 0 ; :: thesis: contradiction
then f . r in {0} by TARSKI:def 1;
hence contradiction by A2, A3, A5, FUNCT_1:def 7; :: thesis: verum

end;

let s1 be Real_Sequence; :: thesis: ( rng s1 c= dom ((f ^) | X) & s1 is convergent & lim s1 = r implies ( ((f ^) | X) /* s1 is convergent & lim (((f ^) | X) /* s1) = ((f ^) | X) . r ) )
assume that
A9: rng s1 c= dom ((f ^) | X) and
A10: ( s1 is convergent & lim s1 = r ) ; :: thesis: ( ((f ^) | X) /* s1 is convergent & lim (((f ^) | X) /* s1) = ((f ^) | X) . r )
rng s1 c= (dom (f ^)) /\ X by A9, RELAT_1:61;
then A11: rng s1 c= dom (f | X) by A3, RELAT_1:61;
then A12: (f | X) /* s1 is convergent by A7, A10;

now :: thesis: for n being Nat holds ((f | X) /* s1) . n <> 0

let n be Nat; :: thesis: ((f | X) /* s1) . n <> 0
A13: s1 . n in rng s1 by VALUED_0:28;
( rng s1 c= (dom f) /\ X & (dom f) /\ X c= dom f ) by A3, A9, RELAT_1:61, XBOOLE_1:17;
then A14: rng s1 c= dom f ;

A15: now :: thesis: not f . (s1 . n) = 0

assume f . (s1 . n) = 0 ; :: thesis: contradiction
then f . (s1 . n) in {0} by TARSKI:def 1;
hence contradiction by A2, A14, A13, FUNCT_1:def 7; :: thesis: verum

end;

n in NAT by ORDINAL1:def 12;
then ((f | X) /* s1) . n = (f | X) . (s1 . n) by A11, FUNCT_2:108
.= f . (s1 . n) by A11, A13, FUNCT_1:47 ;
hence ((f | X) /* s1) . n <> 0 by A15; :: thesis: verum

end;

then A16: (f | X) /* s1 is non-zero by SEQ_1:5;

now :: thesis: for n being Element of NAT holds (((f ^) | X) /* s1) . n = (((f | X) /* s1) ") . n

let n be Element of NAT ; :: thesis: (((f ^) | X) /* s1) . n = (((f | X) /* s1) ") . n
A17: s1 . n in rng s1 by VALUED_0:28;
then s1 . n in dom ((f ^) | X) by A9;
then s1 . n in (dom (f ^)) /\ X by RELAT_1:61;
then A18: s1 . n in dom (f ^) by XBOOLE_0:def 4;
thus (((f ^) | X) /* s1) . n = ((f ^) | X) . (s1 . n) by A9, FUNCT_2:108
.= (f ^) . (s1 . n) by A9, A17, FUNCT_1:47
.= (f . (s1 . n)) " by A18, RFUNCT_1:def 2
.= ((f | X) . (s1 . n)) " by A11, A17, FUNCT_1:47
.= (((f | X) /* s1) . n) " by A11, FUNCT_2:108
.= (((f | X) /* s1) ") . n by VALUED_1:10 ; :: thesis: verum

end;

then A19: ((f ^) | X) /* s1 = ((f | X) /* s1) " by FUNCT_2:63;
A20: (f | X) . r = f . r by A6, FUNCT_1:47;
then lim ((f | X) /* s1) <> 0 by A7, A10, A11, A8;
hence ((f ^) | X) /* s1 is convergent by A12, A16, A19, SEQ_2:21; :: thesis: lim (((f ^) | X) /* s1) = ((f ^) | X) . r
(f | X) . r = lim ((f | X) /* s1) by A7, A10, A11;
hence lim (((f ^) | X) /* s1) = ((f | X) . r) " by A12, A20, A8, A16, A19, SEQ_2:22
.= (f . r) " by A6, FUNCT_1:47
.= (f ^) . r by A5, RFUNCT_1:def 2
.= ((f ^) | X) . r by A4, FUNCT_1:47 ;
:: thesis: verum

end;

hence (f ^) | X is_continuous_in r ; :: thesis: verum