reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;
reserve n for Nat,
  p,q,q1,q2 for Point of TOP-REAL 2,
  r,s1,s2,t1,t2 for Real,
  x,y for Point of Euclid 2;

theorem Th31:
  y = |[0,0]| implies Ball(y,r) = {q : |.q.| < r }
proof
  set X = { q : |.|[0,0]|-q.| < r }, Y = {q : |.q.| < r };
A1: X c= Y
  proof
    let u be object;
    assume u in X;
    then consider q such that
A2: u = q & |.|[0,0]|-q.| < r;
    |.|[0,0]|-q.| = |.q-|[0,0]|.| by TOPRNS_1:27
      .= |.q.| by EUCLID:54,RLVECT_1:13;
    hence thesis by A2;
  end;
A3: Y c= X
  proof
    let u be object;
    assume u in Y;
    then consider q such that
A4: u = q & |.q.| < r;
    |.|[0,0]|-q.| = |.q-|[0,0]|.| by TOPRNS_1:27
      .= |.q.| by EUCLID:54,RLVECT_1:13;
    hence thesis by A4;
  end;
  assume y = |[0,0]|;
  hence Ball(y,r) = { q : |.|[0,0]|-q.| < r } by JGRAPH_2:2
    .= {q : |.q.| < r } by A1,A3;
end;
