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
  |.x1.| - |.x2.| <= |.x1 - x2.|
proof
  reconsider R1=x1,R2=x2 as Element of n-tuples_on REAL;
  x1 = R1 - R2 + R2 by RVSUM_1:43;
  then |.x1.| <= |.x1 - x2.| + |.x2.| by Th9;
  hence thesis by XREAL_1:20;
end;
