reserve k,j,n for Nat,
  r for Real;
reserve x,x1,x2,y for Element of REAL n;
reserve f for real-valued FinSequence;

theorem Th13:
  |.x1 - x2.| = 0 iff x1 = x2
proof
  reconsider R1=x1,R2=x2 as Element of n-tuples_on REAL;
  thus |.x1 - x2.| = 0 implies x1 = x2
  proof
    assume |.x1 - x2.| = 0;
    then R1 - R2 = 0*n by Th5
      .= n |-> 0;
    hence thesis by RVSUM_1:38;
  end;
  assume x1 = x2;
  then R1 - R2 = 0*n by RVSUM_1:37;
  hence thesis by Th4;
end;
