let X be RealUnitarySpace; :: thesis: for x, z being Point of X
for r being Real holds
( z in Sphere (x,r) iff ||.(x - z).|| = r )
let x, z be Point of X; :: thesis: for r being Real holds
( z in Sphere (x,r) iff ||.(x - z).|| = r )
let r be Real; :: thesis: ( z in Sphere (x,r) iff ||.(x - z).|| = r )
thus ( z in Sphere (x,r) implies ||.(x - z).|| = r ) :: thesis: ( ||.(x - z).|| = r implies z in Sphere (x,r) )

proof

assume z in Sphere (x,r) ; :: thesis: ||.(x - z).|| = r
then ex y being Point of X st
( z = y & ||.(x - y).|| = r ) ;
hence ||.(x - z).|| = r ; :: thesis: verum

end;

assume ||.(x - z).|| = r ; :: thesis: z in Sphere (x,r)
hence z in Sphere (x,r) ; :: thesis: verum