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 Th14:
  Gen x,y implies Orte(x,y,Orte(x,y,u)) = -u
proof
  assume
A1: Gen x,y;
  hence Orte(x,y,Orte(x,y,u))=(-pr1(x,y,u))*x+
  (-pr1(x,y,pr2(x,y,u)*x+(-pr1(x,y,u))*y))*y by GEOMTRAP:def 5
    .=(-pr1(x,y,u))*x+(-pr2(x,y,u))*y by A1,GEOMTRAP:def 4
    .=pr1(x,y,u)*(-x)+(-pr2(x,y,u))*y by RLVECT_1:24
    .=-(pr1(x,y,u)*x)+(-pr2(x,y,u))*y by RLVECT_1:25
    .=-(pr1(x,y,u)*x)+pr2(x,y,u)*(-y) by RLVECT_1:24
    .=-(pr1(x,y,u)*x)+(-(pr2(x,y,u)*y)) by RLVECT_1:25
    .=-(pr1(x,y,u)*x+pr2(x,y,u)*y) by RLVECT_1:31
    .=-u by A1,Lm5;
end;
