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

theorem
  Gen w,y implies for u,u9,u1,u2,v1,t1,t2,w1 being VECTOR of V st u<>u9
  & u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y & u,u9,v1,w1
  are_DTr_wrt w,y & v1=u1#u2 holds w1 = t1#t2
proof
  assume
A1: Gen w,y;
  let u,u9,u1,u2,v1,t1,t2,w1 be VECTOR of V such that
A2: u<>u9 and
A3: u,u9,u1,t1 are_DTr_wrt w,y & u,u9,u2,t2 are_DTr_wrt w,y and
A4: u,u9,v1,w1 are_DTr_wrt w,y and
A5: v1=u1#u2;
  set G = PProJ(w,y,u-u9,u+u9), H = PProJ(w,y,u-u9,u1), W = PProJ(w,y,u-u9,u2)
  , I = PProJ(w,y,u-u9,v1), N = PProJ(w,y,u-u9,u-u9);
  set A1 = (G - 2*H)*N", A2 = (G - 2*W)*N", A3 = (G - 2*I)*N";
A6: H+W = PProJ(w,y,u-u9,u1+u2) by A1,Th30
    .= PProJ(w,y,u-u9,v1+v1) by A5,Def2
    .= I+I by A1,Th30;
  v1+A3*(u-u9) =w1 by A1,A2,A4,Th36;
  then
A7: w1+w1 = A3*(u-u9)+(v1 + (v1+A3*(u-u9))) by RLVECT_1:def 3
    .= A3*(u-u9)+((v1 + v1)+A3*(u-u9)) by RLVECT_1:def 3
    .= (v1 + v1)+(A3*(u-u9)+A3*(u-u9)) by RLVECT_1:def 3
    .= (v1 + v1)+(A3+A3)*(u-u9) by RLVECT_1:def 6;
  u1+A1*(u-u9)=t1 & u2+A2*(u-u9)=t2 by A1,A2,A3,Th36;
  then
A8: t1+t2 = A1*(u-u9)+(u1 + (u2+A2*(u-u9))) by RLVECT_1:def 3
    .= A1*(u-u9)+((u1 + u2)+A2*(u-u9)) by RLVECT_1:def 3
    .= (u1 + u2)+(A1*(u-u9)+A2*(u-u9)) by RLVECT_1:def 3
    .= (u1 + u2)+(A1+A2)*(u-u9) by RLVECT_1:def 6
    .= (v1 + v1)+(A1+A2)*(u-u9) by A5,Def2;
  set vv = (G - 2*I) + (G - 2*I);
  A1+A2 = ((G - 2*H) + (G - 2*W))*N" .= vv*N" by A6
    .=A3+A3;
  hence thesis by A8,A7,Def2;
end;
