let S, T be TopSpace; :: thesis: for Y being non empty TopSpace
for A being Subset of T
for f being Function of [:S,T:],Y
for g being Function of [:S,(T | A):],Y st g = f | [:the carrier of S,A:] & f is continuous holds
g is continuous
let Y be non empty TopSpace; :: thesis: for A being Subset of T
for f being Function of [:S,T:],Y
for g being Function of [:S,(T | A):],Y st g = f | [:the carrier of S,A:] & f is continuous holds
g is continuous
let A be Subset of T; :: thesis: for f being Function of [:S,T:],Y
for g being Function of [:S,(T | A):],Y st g = f | [:the carrier of S,A:] & f is continuous holds
g is continuous
let f be Function of [:S,T:],Y; :: thesis: for g being Function of [:S,(T | A):],Y st g = f | [:the carrier of S,A:] & f is continuous holds
g is continuous
let g be Function of [:S,(T | A):],Y; :: thesis: ( g = f | [:the carrier of S,A:] & f is continuous implies g is continuous )
assume that
A1: g = f | [:the carrier of S,A:] and
A2: f is continuous ; :: thesis: g is continuous
set SS = TopStruct(# the carrier of S,the topology of S #);
TopStruct(# the carrier of [:S,T:],the topology of [:S,T:] #) = [:TopStruct(# the carrier of S,the topology of S #),TopStruct(# the carrier of T,the topology of T #):] by Th14;
then A6: TopStruct(# the carrier of [:TopStruct(# the carrier of S,the topology of S #),T:],the topology of [:TopStruct(# the carrier of S,the topology of S #),T:] #) = TopStruct(# the carrier of [:S,T:],the topology of [:S,T:] #) by Th14;
A7: [:TopStruct(# the carrier of S,the topology of S #),(T | A):] = [:(TopStruct(# the carrier of S,the topology of S #) | ([#] TopStruct(# the carrier of S,the topology of S #))),(T | A):] by TSEP_1:3
.= [:TopStruct(# the carrier of S,the topology of S #),T:] | [:([#] TopStruct(# the carrier of S,the topology of S #)),A:] by BORSUK_3:26 ;
TopStruct(# the carrier of [:TopStruct(# the carrier of S,the topology of S #),(T | A):],the topology of [:TopStruct(# the carrier of S,the topology of S #),(T | A):] #) = TopStruct(# the carrier of [:S,(T | A):],the topology of [:S,(T | A):] #) by Th14;
hence g is continuous by A1, A2, A6, A7, TOPMETR:10; :: thesis: verum