reserve a,b,s,t,u,lambda for Real,
  n for Nat;
reserve x,x1,x2,x3,y1,y2 for Element of REAL n;

theorem Th18: :: EUCLID_2:17
  for x being Element of REAL n holds |(x, 0*n)| = 0
proof
  let x be Element of REAL n;
  |(x,0*n)| = 1/4*((|.(x+0*n).|)^2-(|.(x-0*n).|)^2) by EUCLID_2:3
    .= 1/4*((|.x.|)^2-(|.(x-0*n).|)^2) by Th1
    .= 1/4*((|.x.|)^2-(|.x.|)^2) by RVSUM_1:32
    .= 1/4*0;
  hence thesis;
end;
