
theorem Thm37:
  for A,B,C be Point of TOP-REAL 2 st A,B,C are_mutually_distinct &
  C in LSeg(A,B) holds |.A-B.| = |.A-C.| + |.C-B.|
  proof
    let A,B,C be Point of TOP-REAL 2 such that
A1: A,B,C are_mutually_distinct and
A2: C in LSeg(A,B);
    |.B-A.|^2
    =|.A-C.|^2+|.B-C.|^2 - 2 * |.A-C.| * |.B-C.| * (cos angle (A,C,B))
    by EUCLID_6:7
    .= |.A-C.|^2+|.B-C.|^2 - 2 * |.A-C.| * |.B-C.| * (-1)
    by A1,A2,SIN_COS:77,EUCLID_6:8
    .= |.A-C.|^2+ 2 * |.A-C.| * |.B-C.| + |.B-C.|^2
    .= (|.A-C.|+|.B-C.|)^2 by SQUARE_1:4; then
A3: |.B-A.|=|.A-C.|+|.B-C.| or |.B-A.|=-(|.A-C.|+|.B-C.|) by SQUARE_1:40;
    |. B - A .| >= 0 & |. A - C .| >= 0 & |. B - C .| >= 0 &
    not |. B - A .| = 0 & not |. A - C .| = 0 & not |. B - C .| = 0
    by A1,EUCLID_6:42;
    then |.A-B.|=|.A-C.|+|.B-C.| by A3,EUCLID_6:43;
    hence thesis by EUCLID_6:43;
  end;
