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 Th45:
  x _|_ y implies x, y are_lindependent2
proof
  assume
A1: x _|_ y;
  then x, y are_orthogonal;
  then
A2: |(x,y)|=0 by RVSUM_1:def 17;
  x <> 0*n by A1;
  then |(x,x)|<>0 by EUCLID_4:17;
  then
A3: |(x,x)|>0 by RVSUM_1:119;
  y <> 0*n by A1;
  then
A4: |(y,y)|<>0 by EUCLID_4:17;
  then
A5: |(y,y)|>0 by RVSUM_1:119;
  for a,b st a*x+b*y=0*n holds a=0 & b=0
  proof
    let a,b;
    assume a*x+b*y=0*n;
    then
A6: 0 = |(a*x+b*y,a*x+b*y)| by EUCLID_4:17
      .= |(a*x,a*x)|+2*|(a*x,b*y)|+|(b*y,b*y)| by EUCLID_4:32
      .= a*|(x,a*x)|+2*|(a*x,b*y)|+|(b*y,b*y)| by EUCLID_4:21
      .= a*|(x,a*x)|+2*(a*|(x,b*y)|)+|(b*y,b*y)| by EUCLID_4:21
      .= a*|(x,a*x)|+2*a*|(x,b*y)|+b*|(y,b*y)| by EUCLID_4:21
      .= a*(a*|(x,x)|)+2*a*|(x,b*y)|+b*|(y,b*y)| by EUCLID_4:22
      .= a*a*|(x,x)|+2*a*(b*|(x,y)|)+b*|(y,b*y)| by EUCLID_4:22
      .= a*a*|(x,x)|+2*a*b*|(x,y)|+b*(b*|(y,y)|) by EUCLID_4:22
      .= a*a*|(x,x)| + b*b*|(y,y)| by A2;
A7: b*b >= 0 by XREAL_1:63;
A8: a*a = 0
    proof
      assume a*a <> 0;
      then a*a > 0 by XREAL_1:63;
      then a*a*|(x,x)| > 0 by A3,XREAL_1:129;
      hence contradiction by A5,A6,A7;
    end;
    then b*b = 0 by A4,A6,XCMPLX_1:6;
    hence thesis by A8,XCMPLX_1:6;
  end;
  hence thesis;
end;
