theorem
  Bn <> Cn & |(Bn-An,Bn-Cn)| = 0 implies An <> Cn
  proof
    assume that
A1: Bn <> Cn and
A2: |(Bn-An,Bn-Cn)| = 0;
    assume
A3: An = Cn;
    reconsider rB=Bn,rC=Cn as Element of REAL n by EUCLID:22;
    (rB-rC) = 0*n by A2,A3,EUCLID_4:17;
    hence contradiction by A1,EUCLIDLP:9;
  end;
