reserve a,a1,a2,a3,b,b1,b2,b3,r,s,t,u for Real;
reserve n for Nat;
reserve x0,x,x1,x2,x3,y0,y,y1,y2,y3 for Element of REAL n;

theorem
  x1 // x2 implies not x1 _|_ x2
proof
  assume
A1: x1 // x2;
  then consider r such that
A2: x1 = r * x2;
  x2 <> 0*n by A1;
  then
A3: |(x2,x2)| <> 0 by EUCLID_4:17;
  x1 <> 0*n by A1;
  then
A4: r <> 0 by A2,EUCLID_4:3;
  |(x1,x2)| = r*|(x2,x2)| by A2,EUCLID_4:21;
  then |(x1,x2)| <> 0 by A4,A3,XCMPLX_1:6;
  then not x1,x2 are_orthogonal by RVSUM_1:def 17;
  hence thesis;
end;
