let X be non empty TopSpace; :: thesis: for X0 being non empty SubSpace of X
for x0 being Point of X0 holds (modid (X,X0)) | X0 is_continuous_at x0
let X0 be non empty SubSpace of X; :: thesis: for x0 being Point of X0 holds (modid (X,X0)) | X0 is_continuous_at x0
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
reconsider f = (modid (X,A)) | X0 as Function of X0,(X modified_with_respect_to X0) by Def11;
let x0 be Point of X0; :: thesis: (modid (X,X0)) | X0 is_continuous_at x0
A1: (modid (X,A)) | X0 is_continuous_at x0 by Th110;

now

let W be Subset of (X modified_with_respect_to X0); :: thesis: ( W is open & f . x0 in W implies ex V being Subset of X0 st
( V is open & x0 in V & f .: V c= W ) )
reconsider W0 = W as Subset of (X modified_with_respect_to A) by Th114;
assume that
A2: W is open and
A3: f . x0 in W ; :: thesis: ex V being Subset of X0 st
( V is open & x0 in V & f .: V c= W )
W in the topology of (X modified_with_respect_to X0) by A2, PRE_TOPC:def 2;
then W in A -extension_of_the_topology_of X by Th114;
then A4: W0 is open by PRE_TOPC:def 2;
thus ex V being Subset of X0 st
( V is open & x0 in V & f .: V c= W ) :: thesis: verum

proof

consider V being Subset of X0 such that
A5: ( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W0 ) by A1, A3, A4, Th48;
take V ; :: thesis: ( V is open & x0 in V & f .: V c= W )
thus ( V is open & x0 in V & f .: V c= W ) by A5; :: thesis: verum

end;

end;

then f is_continuous_at x0 by Th48;
hence (modid (X,X0)) | X0 is_continuous_at x0 by Def12; :: thesis: verum