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 Th4:
  Gen x,y implies Ortm(x,y,-u) = -Ortm(x,y,u)
proof
  assume
A1: Gen x,y;
  then
A2: -u=-(pr1(x,y,u)*x + pr2(x,y,u)*y) by Lm5
    .=-(pr1(x,y,u)*x) + (-(pr2(x,y,u)*y)) by RLVECT_1:31
    .=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_1:24;
  hence Ortm(x,y,-u)=(-pr1(x,y,u))*x + (-pr2(x,y,-u))*y by A1,Lm6
    .=(-pr1(x,y,u))*x + (-(-pr2(x,y,u)))*y by A1,A2,Lm6
    .=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
    .=-Ortm(x,y,u) by RLVECT_1:31;
end;
