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 Th36:
  Gen w,y implies for p,q,u,v,v9 being VECTOR of V,
   A being Real
st p,q,u,v are_DTr_wrt w,y & p<>q & A = (PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u
  ))*(PProJ(w,y,p-q,p-q))" & v9=u+A*(p-q) holds v=v9
proof
  assume
A1: Gen w,y;
  let p,q,u,v,v9 be VECTOR of V,A be Real;
  assume that
A2: p,q,u,v are_DTr_wrt w,y and
A3: p<>q and
A4: A = (PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u))*(PProJ(w,y,p-q,p-q))" and
A5: v9=u+A*(p-q);
A6: PProJ(w,y,p-q,p-q) <> 0 by A1,A3,Th35;
A7: PProJ(w,y,p-q,A*(p-q)) = ((PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u))*(
  PProJ(w,y,p-q,p-q))")*PProJ (w,y,p-q,p-q) by A1,A4,Th31
    .= (PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u))*((PProJ(w,y,p-q,p-q))"*PProJ
  (w,y,p-q,p-q))
    .= (PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u))*1 by A6,XCMPLX_0:def 7
    .= PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u);
  set s=p#q;
  set X=PProJ(w,y,p-q,(v9#u)-s);
  2*X = 2*(PProJ(w,y,p-q,v9#u)-PProJ(w,y,p-q,p#q)) by A1,Th30
    .= 2*PProJ(w,y,p-q,v9#u)-2*PProJ(w,y,p-q,p#q)
    .= (PProJ(w,y,p-q,v9)+PProJ(w,y,p-q,u))-2*PProJ(w,y,p-q,p#q) by A1,Th34
    .= (PProJ(w,y,p-q,v9)+PProJ(w,y,p-q,u))-(PProJ(w,y,p-q,p)+PProJ (w,y,p-q
  ,q)) by A1,Th34
    .= (PProJ(w,y,p-q,u+A*(p-q)) + PProJ(w,y,p-q,u)) - PProJ (w,y,p-q,p+q)
  by A1,A5,Th30
    .= ((PProJ(w,y,p-q,u) + PProJ(w,y,p-q,A*(p-q))) + PProJ(w,y,p-q,u)) -
  PProJ(w,y,p-q,p+q) by A1,Th30;
  then q,p,p#q,v9#u are_Ort_wrt w,y by A1,A7,Th33;
  then s,v9#u,q,p are_Ort_wrt w,y by Lm5;
  then
A8: s,v9#u,p,q are_Ort_wrt w,y by Lm4;
  set Y = PProJ(w,y,v9-u,(v9#u)-s);
A9: v9-u = A*(p-q) by A5,RLSUB_2:61;
  1*(v9-u) = (u+A*(p-q))-u by A5,RLVECT_1:def 8
    .= A*(p-q) by RLSUB_2:61;
  then q,p // u,v9 or q,p // v9,u by ANALMETR:14;
  then
A10: p,q // u,v9 or p,q // v9,u by ANALOAF:12;
A11: PProJ(w,y,A*(p-q),A*(p-q)) = A*PProJ(w,y,p-q,A*(p-q)) by A1,Th31
    .= A*(((PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u))*(PProJ(w,y,p-q,p-q))")*
  PProJ(w,y,p-q,p-q)) by A1,A4,Th31
    .= A*((PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u))*((PProJ(w,y,p-q,p-q))"*
  PProJ(w,y,p-q,p-q)))
    .= A*((PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u))*1) by A6,XCMPLX_0:def 7
    .= A*(PProJ(w,y,p-q,p+q) - 2*PProJ(w,y,p-q,u));
  2*Y = 2*(PProJ(w,y,v9-u,v9#u)-PProJ(w,y,v9-u,p#q)) by A1,Th30
    .= 2*PProJ(w,y,v9-u,v9#u)-2*PProJ(w,y,v9-u,p#q)
    .= (PProJ(w,y,v9-u,v9)+PProJ(w,y,v9-u,u))-2*PProJ(w,y,v9-u,p#q) by A1,Th34
    .= (PProJ(w,y,v9-u,v9)+PProJ(w,y,v9-u,u))-(PProJ(w,y,v9-u,p)+PProJ (w,y,
  v9-u,q)) by A1,Th34
    .= (PProJ(w,y,v9-u,u+A*(p-q)) + PProJ(w,y,v9-u,u)) - PProJ (w,y,v9-u,p+q
  ) by A1,A5,Th30
    .= ((PProJ(w,y,v9-u,u) + PProJ(w,y,v9-u,A*(p-q))) + PProJ(w,y,v9-u,u)) -
  PProJ (w,y,v9-u,p+q) by A1,Th30;
  then
A12: 2*Y = ((PProJ(w,y,A*(p-q),u) + (A*PProJ(w,y,p-q,p+q) - A*(2*PProJ(w,y,p
  -q,u)))) + PProJ(w,y,A*(p-q),u)) - PProJ(w,y,A*(p-q),p+q) by A9,A11
    .= ((PProJ(w,y,A*(p-q),u) + (PProJ(w,y,A*(p-q),p+q) - 2*(A*PProJ(w,y,p-q
  ,u)))) + PProJ(w,y,A*(p-q),u)) - PProJ(w,y,A*(p-q),p+q) by A1,Th31
    .= ((PProJ(w,y,A*(p-q),u) + (PProJ(w,y,A*(p-q),p+q) - 2*PProJ (w,y,A*(p-
  q),u))) + PProJ(w,y,A*(p-q),u)) - PProJ(w,y,A*(p-q),p+q) by A1,Th31
    .= 0;
  then u,v9,s,v9#u are_Ort_wrt w,y by A1,Th33;
  then s,v9#u,u,v9 are_Ort_wrt w,y by Lm5;
  then s,v9#u,v9,u are_Ort_wrt w,y by Lm4;
  then p,q,s,u#v9 are_Ort_wrt w,y & u,v9,s,u#v9 are_Ort_wrt w,y & p,q,s,v9#u
  are_Ort_wrt w,y & v9,u,s,v9#u are_Ort_wrt w,y by A1,A8,A12,Lm5,Th33;
  then p,q,u,v9 are_DTr_wrt w,y or p,q,v9,u are_DTr_wrt w,y by A10;
  hence thesis by A1,A2,A3,Th25;
end;
