let X be non empty TopSpace; :: thesis: for A being Subset of X
for X0 being non empty SubSpace of X st the carrier of X0 = A holds
for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0

let A be Subset of X; :: thesis: for X0 being non empty SubSpace of X st the carrier of X0 = A holds
for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0

let X0 be non empty SubSpace of X; :: thesis: ( the carrier of X0 = A implies for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0 )
assume A1: the carrier of X0 = A ; :: thesis: for x0 being Point of X0 holds (modid (X,A)) | X0 is_continuous_at x0
let x0 be Point of X0; :: thesis: (modid (X,A)) | X0 is_continuous_at x0
now :: thesis: for W being Subset of () st W is open & ((modid (X,A)) | X0) . x0 in W holds
ex V being Subset of X0 st
( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W )
( x0 in the carrier of X0 & the carrier of X0 c= the carrier of X ) by BORSUK_1:1;
then reconsider x = x0 as Point of X ;
let W be Subset of (); :: thesis: ( W is open & ((modid (X,A)) | X0) . x0 in W implies ex V being Subset of X0 st
( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W ) )

assume that
A2: W is open and
A3: ((modid (X,A)) | X0) . x0 in W ; :: thesis: ex V being Subset of X0 st
( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W )

consider H, G being Subset of X such that
A4: W = H \/ (G /\ A) and
A5: ( H in the topology of X & G in the topology of X ) by A2;
reconsider H = H, G = G as Subset of X ;
A6: (H /\ A) \/ (G /\ A) c= W by ;
((modid (X,A)) | X0) . x0 = (id the carrier of X) . x by FUNCT_1:49
.= x ;
then ( x in H or x in G /\ A ) by ;
then ( x in H /\ A or x in G /\ A ) by ;
then A7: x in (H /\ A) \/ (G /\ A) by XBOOLE_0:def 3;
A8: ((modid (X,A)) | X0) .: ((H \/ G) /\ A) = (id the carrier of X) .: ((H \/ G) /\ A) by
.= (H \/ G) /\ A by FUNCT_1:92 ;
thus ex V being Subset of X0 st
( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W ) :: thesis: verum
proof
reconsider V = (H \/ G) /\ A as Subset of X0 by ;
take V ; :: thesis: ( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W )
( H is open & G is open ) by A5;
then A9: H \/ G is open ;
V = (H \/ G) /\ ([#] X0) by A1;
hence ( V is open & x0 in V & ((modid (X,A)) | X0) .: V c= W ) by ; :: thesis: verum
end;
end;
hence (modid (X,A)) | X0 is_continuous_at x0 by Th43; :: thesis: verum