theorem Th12:
  An <> Bn & Cn = r * An + (1-r) * Bn & Cn = Bn implies r = 0
  proof
    assume that
A1: An <> Bn and
A2: Cn = r * An + (1-r) * Bn and
A3: Cn = Bn;
    reconsider rA = An, rB = Bn, rC = Cn as Element of REAL n by EUCLID:22;
    rC = r * rA + (1 * rB - r * rB) by A2,EUCLIDLP:11
      .= r * rA + (-r * rB) + 1 * rB by RVSUM_1:15
      .= r * rA - r * rB + 1 * rB;
    then 0*n + rB = r * rA - r * rB + 1 * rB by A3,EUCLID_4:1
                 .= (r * rA - r * rB) + rB by EUCLID_4:3;
    then 0*n = r * rA - r * rB by RVSUM_1:25;
    then r * (rA-rB) = 0*n by EUCLIDLP:12;
    then r = 0 or rA-rB=0*n by EUCLID_4:5;
    hence thesis by A1,EUCLIDLP:9;
  end;
