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 X1 is open SubSpace of X0 holds

for x0 being Point of X0

for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let X0, X1 be non empty SubSpace of X; :: thesis: for g being Function of X0,Y st X1 is open SubSpace of X0 holds

for x0 being Point of X0

for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let g be Function of X0,Y; :: thesis: ( X1 is open SubSpace of X0 implies for x0 being Point of X0

for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )

assume A1: X1 is open SubSpace of X0 ; :: thesis: for x0 being Point of X0

for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let x0 be Point of X0; :: thesis: for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let x1 be Point of X1; :: thesis: ( x0 = x1 implies ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )

assume A2: x0 = x1 ; :: thesis: ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

hence ( g is_continuous_at x0 implies g | X1 is_continuous_at x1 ) by A1, Th74; :: thesis: ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 )

thus ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 ) :: thesis: verum

for g being Function of X0,Y st X1 is open SubSpace of X0 holds

for x0 being Point of X0

for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let X0, X1 be non empty SubSpace of X; :: thesis: for g being Function of X0,Y st X1 is open SubSpace of X0 holds

for x0 being Point of X0

for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let g be Function of X0,Y; :: thesis: ( X1 is open SubSpace of X0 implies for x0 being Point of X0

for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )

assume A1: X1 is open SubSpace of X0 ; :: thesis: for x0 being Point of X0

for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let x0 be Point of X0; :: thesis: for x1 being Point of X1 st x0 = x1 holds

( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

let x1 be Point of X1; :: thesis: ( x0 = x1 implies ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 ) )

assume A2: x0 = x1 ; :: thesis: ( g is_continuous_at x0 iff g | X1 is_continuous_at x1 )

hence ( g is_continuous_at x0 implies g | X1 is_continuous_at x1 ) by A1, Th74; :: thesis: ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 )

thus ( g | X1 is_continuous_at x1 implies g is_continuous_at x0 ) :: thesis: verum

proof

reconsider A = the carrier of X1 as Subset of X0 by A1, TSEP_1:1;

assume A3: g | X1 is_continuous_at x1 ; :: thesis: g is_continuous_at x0

A is open by A1, TSEP_1:16;

hence g is_continuous_at x0 by A1, A2, A3, Th77; :: thesis: verum

end;assume A3: g | X1 is_continuous_at x1 ; :: thesis: g is_continuous_at x0

A is open by A1, TSEP_1:16;

hence g is_continuous_at x0 by A1, A2, A3, Th77; :: thesis: verum