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 Th10:
  Gen x,y implies Orte(x,y,u+v)=Orte(x,y,u) + Orte(x,y,v)
proof
  assume
A1: Gen x,y;
  hence Orte(x,y,u+v)=(pr2(x,y,u+v))*x + (-(pr1(x,y,u)+pr1(x,y,v)))*y by Lm7
    .=(pr2(x,y,u)+pr2(x,y,v))*x + (-(pr1(x,y,u)+pr1(x,y,v)))*y by A1,Lm7
    .=pr2(x,y,u)*x + pr2(x,y,v)*x + (-(pr1(x,y,u)+pr1(x,y,v)))*y
  by RLVECT_1:def 6
    .=pr2(x,y,u)*x + pr2(x,y,v)*x + (pr1(x,y,u)+pr1(x,y,v))*(-y)
  by RLVECT_1:24
    .=pr2(x,y,u)*x + pr2(x,y,v)*x + (-((pr1(x,y,u)+pr1(x,y,v))*y))
  by RLVECT_1:25
    .=pr2(x,y,u)*x + pr2(x,y,v)*x + (-(pr1(x,y,u)*y +pr1(x,y,v)*y))
  by RLVECT_1:def 6
    .=pr2(x,y,u)*x + pr2(x,y,v)*x + (-(pr1(x,y,u)*y) +(-(pr1(x,y,v)*y)))
  by RLVECT_1:31
    .=pr2(x,y,u)*x + (pr2(x,y,v)*x + (-(pr1(x,y,u)*y) +(-(pr1(x,y,v)*y))))
  by RLVECT_1:def 3
    .=pr2(x,y,u)*x + (-(pr1(x,y,u)*y) + (pr2(x,y,v)*x +(-(pr1(x,y,v)*y))))
  by RLVECT_1:def 3
    .=pr2(x,y,u)*x + (-(pr1(x,y,u)*y)) + (pr2(x,y,v)*x +(-(pr1(x,y,v)*y)))
  by RLVECT_1:def 3
    .=pr2(x,y,u)*x + (pr1(x,y,u)*(-y)) + (pr2(x,y,v)*x +(-(pr1(x,y,v)*y)))
  by RLVECT_1:25
    .=pr2(x,y,u)*x + (pr1(x,y,u)*(-y)) + (pr2(x,y,v)*x +(pr1(x,y,v)*(-y)))
  by RLVECT_1:25
    .=pr2(x,y,u)*x + ((-pr1(x,y,u))*y) + (pr2(x,y,v)*x +(pr1(x,y,v)*(-y)))
  by RLVECT_1:24
    .=Orte(x,y,u) + Orte(x,y,v) by RLVECT_1:24;
end;
