let X, X1 be set ; :: thesis: for f being PartFunc of COMPLEX,COMPLEX st f is_continuous_on X & X1 c= X holds
f is_continuous_on X1
let f be PartFunc of COMPLEX,COMPLEX; :: thesis: ( f is_continuous_on X & X1 c= X implies f is_continuous_on X1 )
assume that
A1: f is_continuous_on X and
A2: X1 c= X ; :: thesis: f is_continuous_on X1
X c= dom f by A1;
hence A3: X1 c= dom f by A2; :: according to CFCONT_1:def 2 :: thesis: for x0 being Complex st x0 in X1 holds
f | X1 is_continuous_in x0
let g be Complex; :: thesis: ( g in X1 implies f | X1 is_continuous_in g )
assume A4: g in X1 ; :: thesis: f | X1 is_continuous_in g
then A5: f | X is_continuous_in g by A1, A2;
thus f | X1 is_continuous_in g :: thesis: verum
proof
(dom f) /\ X1 c= (dom f) /\ X by A2, XBOOLE_1:26;
then dom (f | X1) c= (dom f) /\ X by RELAT_1:61;
then A6: dom (f | X1) c= dom (f | X) by RELAT_1:61;
g in (dom f) /\ X1 by A3, A4, XBOOLE_0:def 4;
hence A7: g in dom (f | X1) by RELAT_1:61; :: according to CFCONT_1:def 1 :: thesis: for s1 being Complex_Sequence st rng s1 c= dom (f | X1) & s1 is convergent & lim s1 = g holds
( (f | X1) /* s1 is convergent & (f | X1) /. g = lim ((f | X1) /* s1) )
then A8: (f | X) /. g = f /. g by A6, PARTFUN2:15
.= (f | X1) /. g by A7, PARTFUN2:15 ;
let s1 be Complex_Sequence; :: thesis: ( rng s1 c= dom (f | X1) & s1 is convergent & lim s1 = g implies ( (f | X1) /* s1 is convergent & (f | X1) /. g = lim ((f | X1) /* s1) ) )
assume that
A9: rng s1 c= dom (f | X1) and
A10: ( s1 is convergent & lim s1 = g ) ; :: thesis: ( (f | X1) /* s1 is convergent & (f | X1) /. g = lim ((f | X1) /* s1) )
A11: rng s1 c= dom (f | X) by A9, A6;
A12: now :: thesis: for n being Element of NAT holds ((f | X) /* s1) . n = ((f | X1) /* s1) . n
let n be Element of NAT ; :: thesis: ((f | X) /* s1) . n = ((f | X1) /* s1) . n
A13: s1 . n in rng s1 by VALUED_0:28;
thus ((f | X) /* s1) . n = (f | X) /. (s1 . n) by A9, A6, FUNCT_2:109, XBOOLE_1:1
.= f /. (s1 . n) by A11, A13, PARTFUN2:15
.= (f | X1) /. (s1 . n) by A9, A13, PARTFUN2:15
.= ((f | X1) /* s1) . n by A9, FUNCT_2:109 ; :: thesis: verum
end;
( (f | X) /* s1 is convergent & (f | X) /. g = lim ((f | X) /* s1) ) by A5, A10, A11;
hence ( (f | X1) /* s1 is convergent & (f | X1) /. g = lim ((f | X1) /* s1) ) by A8, A12, FUNCT_2:63; :: thesis: verum
end;