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