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 Th40:
  Gen w,y implies for u,u9,u1,u2,v1,v2,t1,t2,w1,w2 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 & u,u9,v2,w2 are_DTr_wrt w,y & u1,u2,v1,v2 are_Ort_wrt w,y
  holds t1,t2,w1,w2 are_Ort_wrt w,y
proof
  assume
A1: Gen w,y;
  let u,u9,u1,u2,v1,v2,t1,t2,w1,w2 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 & u,u9,v2,w2 are_DTr_wrt w,y and
A5: u1,u2,v1,v2 are_Ort_wrt w,y;
  set A1 = (PProJ(w,y,u-u9,u+u9) - 2*PProJ(w,y,u-u9,u1))*(PProJ(w,y,u-u9,u-u9)
)", A2 = (PProJ(w,y,u-u9,u+u9) - 2*PProJ(w,y,u-u9,u2))*(PProJ(w,y,u-u9,u-u9))",
A3 = (PProJ(w,y,u-u9,u+u9) - 2*PProJ(w,y,u-u9,v1))*(PProJ(w,y,u-u9,u-u9))", A4
  = (PProJ(w,y,u-u9,u+u9) - 2*PProJ(w,y,u-u9,v2))*(PProJ(w,y,u-u9,u-u9))";
  (u1+A1*(u-u9))=t1 & (u2+A2*(u-u9))=t2 by A1,A2,A3,Th36;
  then
A6: t2-t1 = (u2 - u1)+(A2-A1)*(u-u9) by A1,Lm21;
A7: t1=u1+A1*(u-u9) & t2=u2+A2*(u-u9) by A1,A2,A3,Th36;
A8: (v1+A3*(u-u9))=w1 & (v2+A4*(u-u9))=w2 by A1,A2,A4,Th36;
  then w2-w1 = (v2 - v1)+(A4-A3)*(u-u9) by A1,Lm21;
  then
A9: PProJ(w,y,t2-t1,w2-w1) =PProJ(w,y,u2-u1,(v2 - v1)+(A4-A3)*(u-u9)) +
  PProJ(w,y,(A2-A1)*(u-u9),(v2 - v1)+(A4-A3)*(u-u9)) by A1,A6,Th30
    .= (PProJ(w,y,u2-u1,v2-v1) + PProJ(w,y,u2-u1,(A4-A3)*(u-u9))) + PProJ(w,
  y,(A2-A1)*(u-u9),(v2 - v1)+(A4-A3)*(u-u9)) by A1,Th30
    .= (0 + PProJ(w,y,u2-u1,(A4-A3)*(u-u9))) + PProJ(w,y,(A2-A1)*(u-u9),(v2
  - v1)+(A4-A3)*(u-u9)) by A1,A5,Th33
    .= PProJ(w,y,u2-u1,(A4-A3)*(u-u9)) + (PProJ(w,y,(A2-A1)*(u-u9),(v2 - v1)
  ) + PProJ (w,y,(A2-A1)*(u-u9),(A4-A3)*(u-u9))) by A1,Th30;
  set aa = (PProJ(w,y,u-u9,u-u9))"*((PProJ(w,y,u-u9,u2-u1)*PProJ (w,y,u-u9,v2-
  v1)));
A10: PProJ(w,y,(A2-A1)*(u-u9),(v2 - v1)) = PProJ(w,y,v2-v1,(A2-A1)*(u-u9))
    .= PProJ(w,y,v2-v1,u-u9)*(A2-A1) by A1,Th31
    .= PProJ(w,y,v2-v1,u-u9)*( ((-2)*PProJ(w,y,u-u9,u2-u1))* ((PProJ(w,y,u-
  u9,u-u9))") ) by A1,A7,Lm21
    .=(-2)*aa;
A11: PProJ(w,y,u-u9,u-u9) <> 0 by A1,A2,Th35;
A12: PProJ(w,y,u2-u1,(A4-A3)*(u-u9)) = PProJ (w,y,u2-u1,u-u9)*(A4-A3) by A1
,Th31
    .= PProJ(w,y,u2-u1,u-u9)*( ((-2)*PProJ(w,y,u-u9,v2-v1))* ((PProJ(w,y,u-
  u9,u-u9))") ) by A1,A8,Lm21
    .=(-2)*aa;
  PProJ(w,y,(A2-A1)*(u-u9),(A4-A3)*(u-u9)) = (A2-A1)*PProJ(w,y,u-u9,(A4-
  A3)*(u-u9)) by A1,Th31
    .= (A2-A1)*((A4-A3)*PProJ(w,y,u-u9,u-u9)) by A1,Th31
    .= (A2-A1)*( (((-2)*PProJ(w,y,u-u9,v2-v1)) * ((PProJ(w,y,u-u9,u-u9))") )
  * PProJ(w,y,u-u9,u-u9) ) by A1,A8,Lm21
    .= (A2-A1)*( ((-2)*PProJ(w,y,u-u9,v2-v1))*(((PProJ(w,y,u-u9,u-u9))" )*
  PProJ(w,y,u-u9,u-u9)) )
    .= (A2-A1)*( ((-2)*PProJ(w,y,u-u9,v2-v1))*1 ) by A11,XCMPLX_0:def 7
    .= ((-2)*(A2-A1))*PProJ(w,y,u-u9,v2-v1)
    .= ((-2)*( ((-2)*PProJ(w,y,u-u9,u2-u1))*((PProJ(w,y,u-u9,u-u9))")) )*
  PProJ(w,y,u-u9,v2-v1) by A1,A7,Lm21
    .=4*aa;
  hence thesis by A1,A9,A12,A10,Th33;
end;
