let r be real number ; :: thesis: for M being MetrStruct
for p, x being Element of M holds
( x in Ball p,r iff ( not M is empty & dist p,x < r ) )
let M be MetrStruct ; :: thesis: for p, x being Element of M holds
( x in Ball p,r iff ( not M is empty & dist p,x < r ) )
let p, x be Element of M; :: thesis: ( x in Ball p,r iff ( not M is empty & dist p,x < r ) )

hereby :: thesis: ( not M is empty & dist p,x < r implies x in Ball p,r )

assume A1: x in Ball p,r ; :: thesis: ( not M is empty & dist p,x < r )
A2: not M is empty by A1;
then reconsider M' = M as non empty MetrStruct ;
reconsider p' = p as Element of M' ;
x in { q where q is Element of M' : dist p',q < r } by A1, Lm6;
then ex q being Element of M st
( x = q & dist p,q < r ) ;
hence ( not M is empty & dist p,x < r ) by A2; :: thesis: verum

end;

assume not M is empty ; :: thesis: ( not dist p,x < r or x in Ball p,r )
then reconsider M' = M as non empty MetrStruct ;
reconsider p' = p as Element of M' ;
assume dist p,x < r ; :: thesis: x in Ball p,r
then x in { q where q is Element of M' : dist p',q < r } ;
hence x in Ball p,r by Lm6; :: thesis: verum