
theorem ThLawOfCosines:
  for a, b, c being POINT of TarskiEuclid2Space holds
    (dist(c,a))^2 = (dist(a,b))^2 + (dist(b,c))^2 - 2 * dist(a,b) *
    dist(b,c) * cos angle(Tn2TR a, Tn2TR b, Tn2TR c)
  proof
    let a, b, c be POINT of TarskiEuclid2Space;
    set ta = |.Tn2TR a - Tn2TR b.|, tb = |. Tn2TR c - Tn2TR b .|,
      tc = |.Tn2TR c - Tn2TR a.|;
    |.Tn2TR a - Tn2TR b.| = dist(a,b) & |. Tn2TR c - Tn2TR b .| = dist(c,b) &
    |.Tn2TR c - Tn2TR a.| = dist(c,a) by ThEquiv;
    hence thesis by EUCLID_6:7;
  end;
