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
  AS is right_transitive & (AS is Euclidean_like or AS is
  Minkowskian_like) implies AS is left_transitive
proof
  assume
A1: AS is right_transitive;
  (for u,u1,v,v1 being Element of AS holds (u,u1 '//' v,v1 implies v,v1
'//' u,u1 ) or for u,u1,v,v1 being Element of AS holds (u,u1 '//' v,v1 implies
v,v1 '//' u1,u )) implies for u,u1,u2,v,v1,v2,w,w1 being Element of AS holds (
u,u1 '//' v,v1 & v,v1 '//' w,w1 & u,u1 '//' u2,v2 implies (u=u1 or v=v1 or u2,
  v2 '//' w,w1 ))
  proof
    assume
A2: for u,u1,v,v1 being Element of AS holds (u,u1 '//' v,v1 implies v,
    v1 '//' u,u1 ) or for u,u1,v,v1 being Element of AS holds (u,u1 '//' v,v1
    implies v,v1 '//' u1,u );
    let u,u1,u2,v,v1,v2,w,w1;
    assume that
A3: u,u1 '//' v,v1 and
A4: v,v1 '//' w,w1 and
A5: u,u1 '//' u2,v2;
A6: now
      assume
A7:   for u,u1,v,v1 being Element of AS holds (u,u1 '//' v,v1 implies
      v,v1 '//' u1,u );
      now
        w,w1 '//' v1,v by A4,A7;
        then
A8:     w1,w '//' v,v1 by Def1;
A9:     u2,v2 '//' u1,u by A5,A7;
        assume that
A10:    u<>u1 and
A11:    v<>v1;
        v,v1 '//' u1,u by A3,A7;
        then w1,w '//' u2,v2 by A1,A10,A11,A8,A9;
        hence thesis by A7;
      end;
      hence thesis;
    end;
    now
      assume
A12:  for u,u1,v,v1 being Element of AS holds (u,u1 '//' v,v1 implies
      v,v1 '//' u,u1 );
      now
A13:    u2,v2 '//' u,u1 by A5,A12;
A14:    w,w1 '//' v,v1 by A4,A12;
        assume that
A15:    u<>u1 and
A16:    v<>v1;
        v,v1 '//' u,u1 by A3,A12;
        then w,w1 '//' u2,v2 by A1,A15,A16,A14,A13;
        hence thesis by A12;
      end;
      hence thesis;
    end;
    hence thesis by A2,A6;
  end;
  hence thesis;
end;
