let X be non empty TopSpace; :: thesis: for X0, X1 being non empty SubSpace of X st X0 misses X1 holds
for x1 being Point of X1 holds (modid X,X0) | X1 is_continuous_at x1
let X0, X1 be non empty SubSpace of X; :: thesis: ( X0 misses X1 implies for x1 being Point of X1 holds (modid X,X0) | X1 is_continuous_at x1 )
assume A1: X0 misses X1 ; :: thesis: for x1 being Point of X1 holds (modid X,X0) | X1 is_continuous_at x1
let x1 be Point of X1; :: thesis: (modid X,X0) | X1 is_continuous_at x1
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
the carrier of X1 misses A by A1, TSEP_1:def 3;
then A2: (modid X,A) | X1 is_continuous_at x1 by Th109;
X modified_with_respect_to A = X modified_with_respect_to X0 by Def10;
then reconsider f = (modid X,A) | X1 as Function of X1,(X modified_with_respect_to X0) ;
f is_continuous_at x1

proof

now

let W be Subset of (X modified_with_respect_to X0); :: thesis: ( W is open & f . x1 in W implies ex V being Subset of X1 st
( V is open & x1 in V & f .: V c= W ) )
assume A3: ( W is open & f . x1 in W ) ; :: thesis: ex V being Subset of X1 st
( V is open & x1 in V & f .: V c= W )
then W in the topology of (X modified_with_respect_to X0) by PRE_TOPC:def 5;
then A4: W in A -extension_of_the_topology_of X by Th114;
reconsider W0 = W as Subset of (X modified_with_respect_to A) by Th114;
A5: W0 is open by A4, PRE_TOPC:def 5;
thus ex V being Subset of X1 st
( V is open & x1 in V & f .: V c= W ) :: thesis: verum

proof

consider V being Subset of X1 such that
A6: ( V is open & x1 in V & ((modid X,A) | X1) .: V c= W0 ) by A2, A3, A5, Th48;
take V ; :: thesis: ( V is open & x1 in V & f .: V c= W )
thus ( V is open & x1 in V & f .: V c= W ) by A6; :: thesis: verum

end;

end;

hence f is_continuous_at x1 by Th48; :: thesis: verum

end;

hence (modid X,X0) | X1 is_continuous_at x1 by Def11; :: thesis: verum