theorem
  Cn = r * An + (1-r) * Bn & r = 1 implies Cn = An
  proof
    assume that
A1: Cn = r * An + (1-r) * Bn and
A2: r = 1;
    reconsider rA=An,rB=Bn,rC=Cn as Element of REAL n by EUCLID:22;
    rC = rA + 0 * rB by A1,A2,EUCLID_4:3;
    then rC = rA + 0*n by EUCLID_4:3;
    hence thesis by EUCLID_4:1;
  end;
