reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;

theorem  Th17:
  Bn <> Cn & Cn <> An & An <> Bn &
  |(Cn-An,Bn-Cn)| is non zero & |(Bn-Cn,An-Bn)| is non zero &
  |(Cn-An,An-Bn)| is non zero &
  r = -(|(Bn,Cn)| -|(Cn,Cn)| -|(An,Bn)|+|(An,Cn)|)/|(Bn-Cn,Bn-Cn)|&
  s = -(|(Cn,An)| -|(An,An)| -|(Bn,Cn)|+|(Bn,An)|)/|(Cn-An,Cn-An)|&
  t = -(|(An,Bn)| -|(Bn,Bn)| -|(Cn,An)|+|(Cn,Bn)|)/|(An-Bn,An-Bn)|
  implies r / (1-r) * s / (1-s) * t / (1-t) = 1
  proof
    assume that
A1: Bn <> Cn and
A2: Cn <> An and
A3: An <> Bn and
A4: |(Cn-An,Bn-Cn)| is non zero & |(Bn-Cn,An-Bn)| is non zero &
    |(Cn-An,An-Bn)| is non zero and
A5: r = -(|(Bn,Cn)| -|(Cn,Cn)| -|(An,Bn)|+|(An,Cn)|)/|(Bn-Cn,Bn-Cn)| and
A6: s = -(|(Cn,An)| -|(An,An)| -|(Bn,Cn)|+|(Bn,An)|)/|(Cn-An,Cn-An)| and
A7: t = -(|(An,Bn)| -|(Bn,Bn)| -|(Cn,An)|+|(Cn,Bn)|)/|(An-Bn,An-Bn)|;
    reconsider rA=An,rB=Bn,rC=Cn as Element of REAL n by EUCLID:22;
A8: rB - rC <> 0*n by A1,EUCLIDLP:9;
A9: rC - rA <> 0*n by A2,EUCLIDLP:9;
A10: rA - rB <> 0*n by A3,EUCLIDLP:9;
    set rBC = |(Bn-Cn,Bn-Cn)|, rCA = |(Cn-An,Cn-An)|,rAB=|(An-Bn,An-Bn)|,
          A = An, B = Bn, C = Cn;
A11: r * rBC = -(|(B,C)| -|(C,C)| -|(A,B)|+|(A,C)|)/rBC*rBC by A5
            .= -(|(B,C)| -|(C,C)| -|(A,B)|+|(A,C)|)
                by A8,EUCLID_4:17,XCMPLX_1:87;
    (r / (1-r)) = (r /(1-r)) * rBC / rBC by A8,EUCLID_4:17,XCMPLX_1:89
               .= r * rBC / (1-r) /rBC
               .= r * rBC / ((1-r)*rBC) by XCMPLX_1:78
               .= (-(|(B,C)| -|(C,C)| -|(A,B)|+|(A,C)|)) /
                  (|(B-C,B-C)|+ |(B,C)| -|(C,C)| -|(A,B)|+|(A,C)|) by A11;
    then
A12: r/(1-r) = (-|(C-A,B-C)|)/(|(B-C,B-C)|
                + (|(B,C)| -|(C,C)| -|(A,B)|+|(A,C)|)) by Th13
            .= (-|(C-A,B-C)|)/(|(B-C,B-C)|+ |(C-A,B-C)|) by Th13
            .= (-|(C-A,B-C)|)/(|(B-C,B-A)|) by Th13;
A13: s * rCA = -(|(C,A)| -|(A,A)| -|(B,C)|+|(B,A)|)/rCA*rCA by A6
            .= -(|(C,A)| -|(A,A)| -|(B,C)|+|(B,A)|)
                 by A9,EUCLID_4:17,XCMPLX_1:87;
    (s / (1-s)) = (s /(1-s)) * rCA / rCA by A9,EUCLID_4:17,XCMPLX_1:89
               .= s * rCA / (1-s) /rCA
               .= s * rCA / ((1-s)*rCA) by XCMPLX_1:78
               .= (-(|(C,A)| -|(A,A)| -|(B,C)|+|(B,A)|)) /
                  (|(C-A,C-A)|+ (|(C,A)| -|(A,A)| -|(B,C)|+|(B,A)|)) by A13;
    then
A14: s/(1-s) = (-|(C-A,A-B)|)/(|(C-A,C-A)|
                + (|(C,A)| -|(A,A)| -|(B,C)|+|(B,A)|)) by Th13
            .= (-|(C-A,A-B)|)/(|(C-A,C-A)|+ |(C-A,A-B)|) by Th13
            .= (-|(C-A,A-B)|)/(|(C-A,C-B)|) by Th13;
A15: t * rAB = -(|(A,B)| -|(B,B)| -|(C,A)|+|(C,B)|)/rAB*rAB by A7
            .= -(|(A,B)| -|(B,B)| -|(C,A)|+|(C,B)|)
                 by A10,EUCLID_4:17,XCMPLX_1:87;
    (t / (1-t)) = (t /(1-t)) * rAB / rAB by A10,EUCLID_4:17,XCMPLX_1:89
               .= t * rAB / (1-t) /rAB
               .= t * rAB / ((1-t)*rAB) by XCMPLX_1:78
               .= (-(|(A,B)| -|(B,B)| -|(C,A)|+|(C,B)|))
                  / (|(A-B,A-B)|+ (|(A,B)| -|(B,B)| -|(C,A)|+|(C,B)|)) by A15;
    then
A16: t/(1-t) = (-|(A-B,B-C)|)/(|(A-B,A-B)|
                 + (|(A,B)| -|(B,B)| -|(C,A)|+|(C,B)|)) by Th13
            .= (-|(A-B,B-C)|)/(|(A-B,A-B)|+ |(A-B,B-C)|) by Th13
            .= (-|(A-B,B-C)|)/(|(A-B,A-C)|) by Th13;
    r/(1-r)*s/(1-s)*t/(1-t) = ((-|(C-A,B-C)|)/(|(B-C,B-A)|))
                                * ((-|(C-A,A-B)|)/(|(C-A,C-B)|))
                                * ((-|(A-B,B-C)|)/(|(A-B,A-C)|)) by A12,A14,A16
                           .= ((-|(C-A,B-C)|)/(-(|(B-C,A-B)|)))
                                * ((-|(C-A,A-B)|)/(|(C-A,C-B)|))
                                * ((-|(A-B,B-C)|)/(|(A-B,A-C)|)) by Th14
                           .= ((-|(C-A,B-C)|)/(-(|(B-C,A-B)|)))
                                * ((-|(C-A,A-B)|)/(-(|(C-A,B-C)|)))
                                * ((-|(A-B,B-C)|)/(|(A-B,A-C)|)) by Th14
                           .= ((-|(C-A,B-C)|)/(-(|(B-C,A-B)|)))
                                * ((-|(C-A,A-B)|)/(-(|(C-A,B-C)|)))
                                * ((-|(B-C,A-B)|)/(-(|(C-A,A-B)|))) by Th14
                           .= 1 by A4,Th3;
    hence thesis;
  end;
