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 Th44:
  x _|_ y0 & y0 // y1 implies x _|_ y1
proof
  assume that
A1: x _|_ y0 and
A2: y0 // y1;
A3: x,y0 are_orthogonal by A1;
  consider r such that
A4: y1 = r * y0 by A2,Def1;
  |(x,y1)| = r*|(x,y0)| by A4,EUCLID_4:22
    .= r * 0 by A3,RVSUM_1:def 17
    .= 0;
  then
A5: x,y1 are_orthogonal by RVSUM_1:def 17;
  x <> 0*n & y1 <> 0*n by A1,A2;
  hence thesis by A5;
end;
