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 Th45:
  for x being Point of Euclid 2 st x = |[a,b]| holds cl_Ball(x,r)
  = closed_inside_of_circle(a,b,r)
proof
  let x be Point of Euclid 2 such that
A1: x = |[a,b]|;
  hereby
    let w be object;
    assume
A2: w in cl_Ball(x,r);
    then reconsider u = w as Point of TOP-REAL 2 by TOPREAL3:8;
    reconsider e = u as Point of Euclid 2 by TOPREAL3:8;
    dist(e,x) = |. u - |[a,b]| .| by A1,JGRAPH_1:28;
    then |. u - |[a,b]| .| <= r by A2,METRIC_1:12;
    hence w in closed_inside_of_circle(a,b,r) by Th42;
  end;
  let w be object;
  assume
A3: w in closed_inside_of_circle(a,b,r);
  then reconsider u = w as Point of TOP-REAL 2;
  reconsider e = u as Point of Euclid 2 by TOPREAL3:8;
  dist(e,x) = |. u - |[a,b]| .| by A1,JGRAPH_1:28;
  then dist(e,x) <= r by A3,Th42;
  hence thesis by METRIC_1:12;
end;
