let X, Y be non empty TopSpace; :: thesis: for X0, X1 being non empty SubSpace of X
for f being Function of X,Y st X1 is SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1

let X0, X1 be non empty SubSpace of X; :: thesis: for f being Function of X,Y st X1 is SubSpace of X0 holds
for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1

let f be Function of X,Y; :: thesis: ( X1 is SubSpace of X0 implies for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1 )

assume A1: X1 is SubSpace of X0 ; :: thesis: for x0 being Point of X0
for x1 being Point of X1 st x0 = x1 & f | X0 is_continuous_at x0 holds
f | X1 is_continuous_at x1

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

let x1 be Point of X1; :: thesis: ( x0 = x1 & f | X0 is_continuous_at x0 implies f | X1 is_continuous_at x1 )
assume A2: x0 = x1 ; :: thesis: ( not f | X0 is_continuous_at x0 or f | X1 is_continuous_at x1 )
assume f | X0 is_continuous_at x0 ; :: thesis: f | X1 is_continuous_at x1
then (f | X0) | X1 is_continuous_at x1 by A1, A2, Th74;
hence f | X1 is_continuous_at x1 by ; :: thesis: verum