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 Th43:
  t in inside_of_circle(a,b,r) iff |. t - |[a,b]| .| < r
proof
A1: inside_of_circle(a,b,r) = {x where x is Point of TOP-REAL 2: |. x - |[a,
  b]| .| < r} by JGRAPH_6:def 6;
  hereby
    assume t in 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;
