reserve i,n,m for Nat;

theorem Th23:
for x0 be Element of REAL m, y0 be Point of REAL-NS m,
  r be Real st x0=y0 holds
  {y where y is Element of REAL m: |.y-x0.| < r}
    = {z where z is Point of REAL-NS m: ||.z-y0.|| < r}
proof
   let x0 be Element of REAL m, y0 be Point of REAL-NS m, r be Real;
   assume A1: x0=y0;
   now let w be object;
    assume w in {y where y is Element of REAL m: |.y-x0.| < r}; then
    consider y be Element of REAL m such that
A2:  y=w & |.y-x0.| < r;
    reconsider z=y as Point of REAL-NS m by REAL_NS1:def 4;
    ||.z-y0.|| < r by A1,A2,REAL_NS1:1,5;
    hence w in {z1 where z1 is Point of REAL-NS m: ||.z1-y0.|| < r} by A2;
   end; then
A3: {y where y is Element of REAL m: |.y-x0.| < r}
     c= {z where z is Point of REAL-NS m: ||.z-y0.|| < r} by TARSKI:def 3;
   now let w be object;
    assume w in {z where z is Point of REAL-NS m: ||.z-y0.|| < r}; then
    consider y be Point of REAL-NS m such that
A4:  y=w & ||.y-y0.|| < r;
    reconsider z=y as Element of REAL m by REAL_NS1:def 4;
    |.z-x0.| < r by A1,A4,REAL_NS1:1,5;
    hence w in {z1 where z1 is Element of REAL m: |.z1-x0.| < r} by A4;
   end; then
   {z where z is Point of REAL-NS m: ||.z-y0.|| < r}
     c= {y where y is Element of REAL m: |.y-x0.| < r} by TARSKI:def 3;
   hence thesis by A3,XBOOLE_0:def 10;
end;
