
theorem Th13:
for z being Point of [:RNS_Real,RNS_Real:], r be Real st 0 < r holds
  ex s be Real, x,y be Real st 0 < s & s < r & z = [x,y]
      & [: ].x-s,x+s.[, ].y-s,y+s.[ :] c= Ball(z,r)
proof
    let z be Point of [:RNS_Real,RNS_Real:], r be Real;
    assume
A1:  0 < r;
    consider xx,yy be Point of RNS_Real such that
A2:  z = [xx,yy] by PRVECT_3:18;
    reconsider x=xx,y=yy as Real;

    consider s being Real such that
A3:  0 < s & s < r & [:Ball (xx,s),Ball (yy,s):] c= Ball (z,r)
       by A1,A2,NDIFF_8:22;
    Ball(xx,s) = ].x-s,x+s.[ & Ball(yy,s) = ].y-s,y+s.[ by Th12;
    hence thesis by A2,A3;
end;
