reserve V for RealLinearSpace,
  u,u1,u2,v,v1,v2,w,w1,x,y for VECTOR of V,
  a,a1,a2,b,b1,b2,c1,c2,n,k1,k2 for Real;

theorem Th9:
  Gen x,y implies Orte(x,y,-v)= -Orte(x,y,v)
proof
  assume
A1: Gen x,y;
  then
A2: -v=-(pr1(x,y,v)*x + pr2(x,y,v)*y) by Lm5
    .=-(pr1(x,y,v)*x) + (-(pr2(x,y,v)*y)) by RLVECT_1:31
    .=pr1(x,y,v)*(-x) + (-(pr2(x,y,v)*y)) by RLVECT_1:25
    .=(-pr1(x,y,v))*x + (-(pr2(x,y,v)*y)) by RLVECT_1:24
    .=(-pr1(x,y,v))*x + pr2(x,y,v)*(-y) by RLVECT_1:25
    .=(-pr1(x,y,v))*x + (-pr2(x,y,v))*y by RLVECT_1:24;
  hence Orte(x,y,-v)=(-pr2(x,y,v))*x + (-pr1(x,y,-v))*y by A1,Lm6
    .=(-pr2(x,y,v))*x + (-(-pr1(x,y,v)))*y by A1,A2,Lm6
    .=pr2(x,y,v)*(-x) + (-(-pr1(x,y,v)))*y by RLVECT_1:24
    .=-(pr2(x,y,v)*x) + (-(-pr1(x,y,v)))*y by RLVECT_1:25
    .=-(pr2(x,y,v)*x) + (-pr1(x,y,v))*(-y) by RLVECT_1:24
    .=-(pr2(x,y,v)*x) + (-((-pr1(x,y,v))*y)) by RLVECT_1:25
    .=-Orte(x,y,v) by RLVECT_1:31;
end;
