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 Th17: :: EUCLID_2:15
  for x being Element of REAL n holds |(x,x)|=0 iff x=0*n
proof
  let x be Element of REAL n;
  thus |(x,x)| = 0 implies x = 0*n
  proof
    assume |(x,x)| = 0;
    then |.x.| = 0;
    hence thesis by EUCLID:8;
  end;
  thus x = 0*n implies |(x,x)| = 0
  proof
    assume x = 0*n;
    then |(x,x)| = 1/4*((|.(0*n+0*n).|)^2-(|.(0*n-0*n).|)^2) by EUCLID_2:3
      .= 1/4*((|.(0*n).|)^2-(|.(0*n-0*n).|)^2) by Th1
      .= 1/4*0 by RVSUM_1:32;
    hence thesis;
  end;
end;
