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

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

assume A1: x in Sphere p,r ; :: thesis: ( not M is empty & dist p,x = r )
then reconsider M9 = M as non empty MetrStruct ;
reconsider p9 = p as Element of M9 ;
x in { q where q is Element of M9 : dist p9,q = r } by A1, Lm7;
then ex q being Element of M st
( x = q & dist p,q = r ) ;
hence ( not M is empty & dist p,x = r ) by A1; :: thesis: verum

end;

assume not M is empty ; :: thesis: ( not dist p,x = r or x in Sphere p,r )
then reconsider M9 = M as non empty MetrStruct ;
reconsider p9 = p as Element of M9 ;
assume dist p,x = r ; :: thesis: x in Sphere p,r
then x in { q where q is Element of M9 : dist p9,q = r } ;
hence x in Sphere p,r by Lm7; :: thesis: verum