
theorem Th12:
for z being Point of RNS_Real, x,r be Real st x = z holds
 Ball(z,r) = ].x-r,x+r.[
proof
    let z be Point of RNS_Real, x,r be Real;
    assume
A1:  x = z;

    for p be object holds p in Ball(z,r) iff p in ]. x-r,x+r .[
    proof
     let p be object;
     hereby assume p in Ball(z,r); then
      consider y be Point of RNS_Real such that
A2:    p = y & ||. z-y .|| < r;
      reconsider u = y as Real;
A3:   u - x = y - z by A1,DUALSP03:4;
      ||. y-z .|| < r by A2,NORMSP_1:7; then
      |. u-x .| < r by A3,EUCLID:def 2;
      hence p in ].x-r,x+r.[ by A2,RCOMP_1:1;
     end;
     assume
A4:   p in ]. x-r, x+r .[; then
     reconsider u=p as Real;
A5:  |. u-x .| < r by A4,RCOMP_1:1;
     reconsider y = u as Point of RNS_Real by XREAL_0:def 1;
     u-x = y-z by A1,DUALSP03:4; then
     ||. y-z .|| < r by A5,EUCLID:def 2; then
     ||. z-y .|| < r by NORMSP_1:7;
     hence p in Ball(z,r);
    end;
    hence thesis by TARSKI:2;
end;
