let M be non empty MetrSpace; :: thesis: for x being Point of (TopSpaceMetr M)
for x9 being Point of M st x = x9 holds
ex f being sequence of (Balls x) st
for n being Element of NAT holds f . n = Ball (x9,(1 / (n + 1)))
let x be Point of (TopSpaceMetr M); :: thesis: for x9 being Point of M st x = x9 holds
ex f being sequence of (Balls x) st
for n being Element of NAT holds f . n = Ball (x9,(1 / (n + 1)))
let x9 be Point of M; :: thesis: ( x = x9 implies ex f being sequence of (Balls x) st
for n being Element of NAT holds f . n = Ball (x9,(1 / (n + 1))) )
assume A1: x = x9 ; :: thesis: ex f being sequence of (Balls x) st
for n being Element of NAT holds f . n = Ball (x9,(1 / (n + 1)))
set B = Balls x;
consider x9 being Point of M such that
A2: ( x9 = x & Balls x = { (Ball (x9,(1 / n))) where n is Nat : n <> 0 } ) by Def1;
defpred S₁[ object , object ] means ex n9 being Element of NAT st
( $1 = n9 & $2 = Ball (x9,(1 / (n9 + 1))) );
A3: for n being object st n in NAT holds
ex O being object st
( O in Balls x & S₁[n,O] )

proof

let n be object ; :: thesis: ( n in NAT implies ex O being object st
( O in Balls x & S₁[n,O] ) )
assume n in NAT ; :: thesis: ex O being object st
( O in Balls x & S₁[n,O] )
then reconsider n = n as Element of NAT ;
take Ball (x9,(1 / (n + 1))) ; :: thesis: ( Ball (x9,(1 / (n + 1))) in Balls x & S₁[n, Ball (x9,(1 / (n + 1)))] )
thus ( Ball (x9,(1 / (n + 1))) in Balls x & S₁[n, Ball (x9,(1 / (n + 1)))] ) by A2; :: thesis: verum

end;

consider f being Function such that
A4: ( dom f = NAT & rng f c= Balls x ) and
A5: for n being object st n in NAT holds
S₁[n,f . n] from FUNCT_1:sch 6(A3);
reconsider f = f as sequence of (Balls x) by A4, FUNCT_2:2;
take f ; :: thesis: for n being Element of NAT holds f . n = Ball (x9,(1 / (n + 1)))
let n be Element of NAT ; :: thesis: f . n = Ball (x9,(1 / (n + 1)))
S₁[n,f . n] by A5;
hence f . n = Ball (x9,(1 / (n + 1))) by A2, A1; :: thesis: verum