reserve n for Nat,
        lambda,lambda2,mu,mu2 for Real,
        x1,x2 for Element of REAL n,
        An,Bn,Cn for Point of TOP-REAL n,
        a for Real;
 reserve Pn,PAn,PBn for Element of REAL n,
         Ln for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;

theorem Th8:
  C in LSeg(A,B) implies |.A-B.| = |.A-C.| + |.C-B.|
  proof
    assume
A1: C in LSeg(A,B);
    per cases;
    suppose A,B,C are_mutually_distinct;
      hence thesis by A1,EUCLID10:57;
    end;
    suppose not A,B,C are_mutually_distinct;
      then per cases;
      suppose A=B;
        then LSeg(A,B) = {A} by RLTOPSP1:70;
        then |.C-B.| = |.A-B.| & |.A-C.| = |.C-C.| & |.C-C.| = 0
                by A1,TARSKI:def 1,EUCLID_6:42;
        hence thesis;
      end;
      suppose C=B;
        then |.C-B.| = |.B-B.| & |.A-C.| = |.A-B.| & |.B-B.| = 0
                by EUCLID_6:42;
        hence thesis;
      end;
      suppose C=A;
        then |.A-C.| = |.C-C.| & |.A-B.| = |.C-B.| & |.C-C.| = 0
                by EUCLID_6:42;
        hence thesis;
      end;
    end;
  end;
