let X, Y, Z be non empty TopSpace; :: thesis: ( the carrier of X = the carrier of Y & the topology of Y c= the topology of X implies for f being Function of X,Z
for g being Function of Y,Z st f = g holds
for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x )

assume that
A1: the carrier of X = the carrier of Y and
A2: the topology of Y c= the topology of X ; :: thesis: for f being Function of X,Z
for g being Function of Y,Z st f = g holds
for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x

let f be Function of X,Z; :: thesis: for g being Function of Y,Z st f = g holds
for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x

let g be Function of Y,Z; :: thesis: ( f = g implies for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x )

assume A3: f = g ; :: thesis: for x being Point of X
for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x

let x be Point of X; :: thesis: for y being Point of Y st x = y & g is_continuous_at y holds
f is_continuous_at x

let y be Point of Y; :: thesis: ( x = y & g is_continuous_at y implies f is_continuous_at x )
assume A4: x = y ; :: thesis: ( not g is_continuous_at y or f is_continuous_at x )
assume A5: g is_continuous_at y ; :: thesis:
for G being Subset of Z st G is open & f . x in G holds
ex H being Subset of X st
( H is open & x in H & f .: H c= G )
proof
let G be Subset of Z; :: thesis: ( G is open & f . x in G implies ex H being Subset of X st
( H is open & x in H & f .: H c= G ) )

assume ( G is open & f . x in G ) ; :: thesis: ex H being Subset of X st
( H is open & x in H & f .: H c= G )

then consider H being Subset of Y such that
A6: H is open and
A7: ( y in H & g .: H c= G ) by A3, A4, A5, Th43;
reconsider F = H as Subset of X by A1;
take F ; :: thesis: ( F is open & x in F & f .: F c= G )
H in the topology of Y by A6;
hence ( F is open & x in F & f .: F c= G ) by A2, A3, A4, A7; :: thesis: verum
end;
hence f is_continuous_at x by Th43; :: thesis: verum