
theorem Thm38:
  for A,B being Point of TOP-REAL 2,
  a,b being Real, r being positive Real st A,B,|[a,b]| are_mutually_distinct &
  A in circle(a,b,r) & B in circle(a,b,r) & |[a,b]| in LSeg(A,B) holds
  |.A-B.| = 2*r
  proof
    let A,B be Point of TOP-REAL 2, a,b be Real,r be positive Real such that
A1: A,B,|[a,b]| are_mutually_distinct and
A2: A in circle(a,b,r) & B in circle(a,b,r) and
A3: |[a,b]| in LSeg(A,B);
A4: circle(a,b,r)={p where p is Point of TOP-REAL 2: |.p-|[a,b]|.|=r}
    by JGRAPH_6:def 5;
    consider JA be Point of TOP-REAL 2 such that
A5: A=JA and
A6: |.JA-|[a,b]|.|=r by A2,A4;
    consider JB be Point of TOP-REAL 2 such that
A7: B=JB and
A8: |.JB-|[a,b]|.|=r by A2,A4;
    |.A-B.| = |.A-|[a,b]|.|+|.|[a,b]|-B.| by A1,A3,Thm37
    .= r + r by A5,A6,A7,A8,EUCLID_6:43;
    hence thesis;
  end;
