let z be Complex; :: thesis: for CNS being ComplexNormSpace
for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
z (#) f is_continuous_on X
let CNS be ComplexNormSpace; :: thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
z (#) f is_continuous_on X
let RNS be RealNormSpace; :: thesis: for X being set
for f being PartFunc of RNS,CNS st f is_continuous_on X holds
z (#) f is_continuous_on X
let X be set ; :: thesis: for f being PartFunc of RNS,CNS st f is_continuous_on X holds
z (#) f is_continuous_on X
let f be PartFunc of RNS,CNS; :: thesis: ( f is_continuous_on X implies z (#) f is_continuous_on X )
assume A1: f is_continuous_on X ; :: thesis: z (#) f is_continuous_on X
then A2: X c= dom f ;
then A3: X c= dom (z (#) f) by VFUNCT_2:def 2;
now :: thesis: for s1 being sequence of RNS st rng s1 c= X & s1 is convergent & lim s1 in X holds
( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) )
let s1 be sequence of RNS; :: thesis: ( rng s1 c= X & s1 is convergent & lim s1 in X implies ( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) ) )
assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X ; :: thesis: ( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) )
A7: f /* s1 is convergent by A1, A4, A5, A6, Th43;
then A8: z * (f /* s1) is convergent by CLVECT_1:116;
f /. (lim s1) = lim (f /* s1) by A1, A4, A5, A6, Th43;
then (z (#) f) /. (lim s1) = z * (lim (f /* s1)) by A3, A6, VFUNCT_2:def 2
.= lim (z * (f /* s1)) by A7, CLVECT_1:122
.= lim ((z (#) f) /* s1) by A2, A4, Th28, XBOOLE_1:1 ;
hence ( (z (#) f) /* s1 is convergent & (z (#) f) /. (lim s1) = lim ((z (#) f) /* s1) ) by A2, A4, A8, Th28, XBOOLE_1:1; :: thesis: verum
end;
hence z (#) f is_continuous_on X by A3, Th43; :: thesis: verum