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 Th19:
  Bn <> Cn & r = -(|(Bn,Cn)| -|(Cn,Cn)| -|(An,Bn)|+|(An,Cn)|)/|(Bn-Cn,Bn-Cn)| &
  r = 1 implies |(Bn-Cn,An-Bn)| = 0
  proof
    assume that
A1: Bn <> Cn and
A2: r = -(|(Bn,Cn)| -|(Cn,Cn)| -|(An,Bn)|+|(An,Cn)|)/|(Bn-Cn,Bn-Cn)| and
A3: r = 1;
    set A = An, B = Bn, C = Cn;
    reconsider rB=B,rC=C as Element of REAL n by EUCLID:22;
A4: rB - rC <> 0*n by A1,EUCLIDLP:9;
    1 * |(B-C,B-C)| = (-(|(C-A,B-C)|)/|(B-C,B-C)|)*|(B-C,B-C)| by A3,A2,Th13;
    then |(B-C,B-C)| = -(|(C-A,B-C)|)/|(B-C,B-C)|*|(B-C,B-C)|;
    then |(B-C,B-C)| = -(|(C-A,B-C)|) by A4,EUCLID_4:17,XCMPLX_1:87;
    hence thesis by Th18;
  end;
