reserve n for Nat,
  a, b, r, w for Real,
  x, y, z for Point of TOP-REAL n,
  e for Point of Euclid n;
reserve V for RealLinearSpace,
        p,q,x for Element of V;
reserve p, q, x for Point of TOP-REAL n;
reserve s, t for Point of TOP-REAL 2;

theorem Th42:
  t in closed_inside_of_circle(a,b,r) iff |. t - |[a,b]| .| <= r
proof
A1: closed_inside_of_circle(a,b,r) = {x where x is Point of TOP-REAL 2: |. x
  - |[a,b]| .| <= r} by JGRAPH_6:def 7;
  hereby
    assume t in closed_inside_of_circle(a,b,r);
    then
    ex x being Point of TOP-REAL 2 st t = x & |. x - |[a,b]| .| <= r by A1;
    hence |. t - |[a,b]| .| <= r;
  end;
  thus thesis by A1;
end;
