reserve V for RealLinearSpace;
reserve x,y for VECTOR of V;
reserve AS for Oriented_Orthogonality_Space;
reserve u,u1,u2,u3,v,v1,v2,v3,w,w1 for Element of AS;

theorem
  for u,v,w being Element of AS holds ex u1 being Element of AS st u1,w
  '//' u,v & u1<>w
proof
  let u,v,w;
  consider u1 such that
A1: u1<>w and
A2: w,u1 '//' v,u by Def1;
  u1,w '//' u,v by A2,Def1;
  hence thesis by A1;
end;
