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 Th18:
  |(Bn-Cn,Bn-Cn)| = - |(Cn-An,Bn-Cn)| implies |(Bn-Cn,An-Bn)| = 0
  proof
    assume
A1: |(Bn-Cn,Bn-Cn)| = - |(Cn-An,Bn-Cn)|;
    reconsider rA=An,rB=Bn,rC=Cn as Element of REAL n by EUCLID:22;
    |(rB-rC,rB-rC)| + |(rB-rC,rC-rA)| =0 by A1;
    then |(rB-rC,(rB-rC)+(rC-rA))| = 0 by EUCLID_4:28;
    then |(rB-rC,(rB-rC)+rC-rA)| = 0 by RVSUM_1:40;
    then |(Bn-Cn,Bn-An)| = 0 by RVSUM_1:43;
    then - |(Bn-Cn,An-Bn)| = 0 by Th14;
    hence thesis;
  end;
