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