reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;

theorem
  for P being Element of TOP-REAL 2,Q being Element of TOP-REAL 2,r being
  Real holds P in Sphere(Q,r) iff P in circle(Q.1,Q.2,r)
  proof
    let P be Element of TOP-REAL 2,
        Q be Element of TOP-REAL 2,
        r be Real;
    hereby
      assume P in Sphere(Q,r);
      then P in Sphere(|[Q`1,Q`2]|,r) by EUCLID:53;
      then P in Sphere(|[Q.1,Q`2]|,r) by EUCLID:def 9;
      then P in Sphere(|[Q.1,Q.2]|,r) by EUCLID:def 10;
      hence P in circle(Q.1,Q.2,r) by TOPREAL9:52;
    end;
    assume P in circle(Q.1,Q.2,r);
    then P in Sphere(|[Q.1,Q.2]|,r) by TOPREAL9:52;
    then P in Sphere(|[Q`1,Q.2]|,r) by EUCLID:def 9;
    then P in Sphere(|[Q`1,Q`2]|,r) by EUCLID:def 10;
    hence thesis by EUCLID:53;
  end;
