
theorem Th5:
for u being Point of [:RNS_Real,RNS_Real,RNS_Real:], r be Real st 0 < r holds
 ex s,x,y,z be Real st 0 < s & s < r & u = [x,y,z]
   & [: ].x-s,x+s.[, ].y-s,y+s.[, ].z-s,z+s.[ :] c= Ball(u,r)
proof
    let u be Point of [:RNS_Real,RNS_Real,RNS_Real:], r be Real;
    assume
A1: 0 < r;
    consider xx,yy,zz be Point of RNS_Real such that
A2: u = [xx,yy,zz] by PRVECT_4:9;
    reconsider u1 = [xx,yy] as Point of [:RNS_Real,RNS_Real:];
    consider s being Real such that
A3: 0 < s & s < r & [:Ball(u1,s), Ball(zz,s):] c= Ball(u,r)
      by NDIFF_8:22,A1,A2;

    consider s1 be Real, x,y be Real such that
A4: 0 < s1 & s1 < s & u1=[x,y]
  & [: ].x-s1,x+s1.[, ].y-s1,y+s1.[ :] c= Ball(u1,s) by A3,MESFUN16:13;

A5: Ball(zz,s1) c= Ball(zz,s) by A4,NDIFF_8:15;

    reconsider z=zz as Real;
    take s1,x,y,z;
    thus 0 < s1 by A4;
    thus s1 < r by A3,A4,XXREAL_0:2;
    thus u = [x,y,z] by A2,A4;

    Ball(zz,s1) = ].z-s1,z+s1.[ by MESFUN16:12; then
    [:[: ].x-s1,x+s1.[, ].y-s1,y+s1.[ :], ].z-s1,z+s1.[ :]
     c= [:Ball(u1,s),Ball(zz,s):] by A4,A5,ZFMISC_1:96;
    hence thesis by A3;
end;
