let X, Y be non empty TopSpace; :: thesis: for X0, X1 being non empty SubSpace of X
for g being Function of X0,Y st TopStruct(# the carrier of X1, the topology of X1 #) = X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0

let X0, X1 be non empty SubSpace of X; :: thesis: for g being Function of X0,Y st TopStruct(# the carrier of X1, the topology of X1 #) = X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0

let g be Function of X0,Y; :: thesis: ( TopStruct(# the carrier of X1, the topology of X1 #) = X0 implies for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0 )

reconsider Y1 = TopStruct(# the carrier of X1, the topology of X1 #) as TopSpace ;
assume A1: TopStruct(# the carrier of X1, the topology of X1 #) = X0 ; :: thesis: for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0

then the carrier of X1 c= the carrier of X0 ;
then reconsider A = the carrier of X1 as Subset of X0 ;
A = [#] X0 by A1;
then A2: A is open ;
Y1 is SubSpace of X0 by ;
then A3: X1 is open SubSpace of X0 by ;
let x0 be Point of X0; :: thesis: for x1 being Point of X1 st x0 = x1 & g | X1 is_continuous_at x1 holds
g is_continuous_at x0

let x1 be Point of X1; :: thesis: ( x0 = x1 & g | X1 is_continuous_at x1 implies g is_continuous_at x0 )
assume A4: x0 = x1 ; :: thesis: ( not g | X1 is_continuous_at x1 or g is_continuous_at x0 )
assume g | X1 is_continuous_at x1 ; :: thesis:
hence g is_continuous_at x0 by A4, A3, Th79; :: thesis: verum