let X be RealNormSpace; :: thesis: for S being sequence of X
for St being sequence of
for x being Point of X
for xt being Point of st S = St & x = xt holds
( St is_convergent_to xt iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - x).|| < r )

let S be sequence of X; :: thesis: for St being sequence of
for x being Point of X
for xt being Point of st S = St & x = xt holds
( St is_convergent_to xt iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - x).|| < r )

let St be sequence of ; :: thesis: for x being Point of X
for xt being Point of st S = St & x = xt holds
( St is_convergent_to xt iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - x).|| < r )

let x be Point of X; :: thesis: for xt being Point of st S = St & x = xt holds
( St is_convergent_to xt iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - x).|| < r )

let xt be Point of ; :: thesis: ( S = St & x = xt implies ( St is_convergent_to xt iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - x).|| < r ) )

reconsider xxt = xt as Point of () by Def4;
assume A1: ( S = St & x = xt ) ; :: thesis: ( St is_convergent_to xt iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - x).|| < r )

the carrier of = the carrier of () by Def4;
then reconsider SSt = St as sequence of () ;
( St is_convergent_to xt iff SSt is_convergent_to xxt ) by Th26;
hence ( St is_convergent_to xt iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - x).|| < r ) by ; :: thesis: verum