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

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

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

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

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

let x be Point of X; :: thesis: ( f is_continuous_at x implies g is_continuous_at x )
assume A4: f is_continuous_at x ; :: thesis:
for G being Subset of Z st G is open & g . x in G holds
ex H being Subset of X st
( H is open & x in H & g .: H c= G )
proof
let G be Subset of Z; :: thesis: ( G is open & g . x in G implies ex H being Subset of X st
( H is open & x in H & g .: H c= G ) )

reconsider F = G as Subset of Y by A1;
assume that
A5: G is open and
A6: g . x in G ; :: thesis: ex H being Subset of X st
( H is open & x in H & g .: H c= G )

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