reserve p,p1,p2,p3,q,q1,q2 for Point of TOP-REAL 2,
  i for Nat,
  lambda for Real;

theorem Th24:
  for r being Real holds inside_of_circle(0,0,r)={p : |.p.|<r} &
  (r>0 implies circle(0,0,r)={p2 : |.p2.|=r}) &
  outside_of_circle(0,0,r)={p3 : |.p3.|>r} &
  closed_inside_of_circle(0,0,r)={q : |.q.|<=r} &
  closed_outside_of_circle(0,0,r)={q2 : |.q2.|>=r}
proof
  let r be Real;
  defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|<r;
  defpred Q[Point of TOP-REAL 2] means |.$1.|<r;
  deffunc F(set)=$1;
A1: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
  inside_of_circle(0,0,r) = {F(p) where p is Point of TOP-REAL 2: P[p]}
    .= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A1
  );
  hence inside_of_circle(0,0,r)={p : |.p.|<r};
  defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|=r;
  defpred Q[Point of TOP-REAL 2] means |.$1.|=r;
A2: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
  hereby
    assume r>0;
    circle(0,0,r)= {F(p): P[p]}
.= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A2);
    hence circle(0,0,r)={p2 : |.p2.|=r};
  end;
  defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|>r;
  defpred Q[Point of TOP-REAL 2] means |.$1.|>r;
A3: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
  outside_of_circle(0,0,r)= {F(p): P[p]}
    .= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A3
  );
  hence outside_of_circle(0,0,r)={p3 : |.p3.|>r};
  defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|<=r;
  defpred Q[Point of TOP-REAL 2] means |.$1.|<=r;
A4: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
  closed_inside_of_circle(0,0,r)= {F(p): P[p]}
    .= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A4
  );
  hence closed_inside_of_circle(0,0,r)={p : |.p.|<=r};
  defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|>=r;
  defpred Q[Point of TOP-REAL 2] means |.$1.|>=r;
A5: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
  closed_outside_of_circle(0,0,r)= {F(p): P[p]}
    .= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A5
  );
  hence thesis;
end;
