
theorem Thm24:
  for A,B,C being Point of TOP-REAL 2,
      a,b,r being Real st A,B,C is_a_triangle &
  A in circle(a,b,r) & B in circle(a,b,r) holds r is positive
  proof
    let A,B,C be Point of TOP-REAL 2, a,b,r be Real such that
A1: A,B,C is_a_triangle and
A2: A in circle(a,b,r) & B in circle(a,b,r);
A3: A,B,C are_mutually_distinct by A1,EUCLID_6:20;
    assume not r is positive;
    then per cases;
    suppose r<0;
      hence contradiction by A2;
    end;
    suppose r=0;
      then A in { |[a,b]| } & B in { |[a,b]| } by A2,Thm23;
      then A = |[a,b]| & B = |[a,b]| by TARSKI:def 1;
      hence contradiction by A3;
    end;
  end;
