reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;
reserve pf for FinSequence of D;
reserve PQR for Matrix of 3,F_Real;
reserve R for Ring;

theorem Th82:
  for N being invertible Matrix of 3,F_Real
  for u,mu being Element of TOP-REAL 3
  for uf being FinSequence of F_Real
  for ut being FinSequence of 1-tuples_on REAL st
  u is non zero & u = uf & ut = N * uf & mu = M2F ut holds
  mu is non zero
  proof
    let N be invertible Matrix of 3,F_Real;
    let u,mu be Element of TOP-REAL 3;
    let uf be FinSequence of F_Real;
    let ut be FinSequence of 1-tuples_on REAL;
    assume that
A1: u is non zero and
A2: u = uf and
A3: ut = N * uf and
A4: mu = M2F ut;
    uf in TOP-REAL 3 by A2; then
A5: uf in REAL 3 by EUCLID:22;
A6: len uf = 3 by A5,EUCLID_8:50;
A7: width <*uf*> = 3 by A6,Th61; then
A8: len (<*uf*>@) = width <*uf*> by MATRIX_0:29
                 .= len uf by MATRIX_0:23; then
A9: len (<*uf*>@) = 3 by A5,EUCLID_8:50;
A10: len <*uf*> = 1 by MATRIX_0:23; then
A11: len (<*uf*>@) = 3 & width(<*uf*>@) = 1 by A7,MATRIX_0:29;
    width N = 3 by MATRIX_0:24;
    then len(N * (<*uf*>@)) = len N &
      width (N * (<*uf*>@)) = width(<*uf*>@) by A11,MATRIX_3:def 4; then
A12: len(N * (<*uf*>@)) = 3 &
      width (N * (<*uf*>@)) = 1 by A10,A7,MATRIX_0:29,MATRIX_0:24;
A13: width N = 3 by MATRIX_0:24
            .= len (<*uf*>@) by A8,A5,EUCLID_8:50;
A14: len ut = len (N * (<*uf*>@)) by A3,LAPLACE:def 9
           .= len N by A13,MATRIX_3:def 4
           .= 3 by MATRIX_0:23;
    assume
A15: mu is zero;
    reconsider MU = M2F ut as FinSequence of REAL;
A16: ut = F2M (MU) by A14,Th69
       .= <* <* 0 *>,<* 0 *>,<* 0 *> *> by A15,A4,EUCLID_5:4,Th65;
A17: (N~) is_reverse_of N by MATRIX_6:def 4;
A18: width (N~) = 3 by MATRIX_0:24;
A19: len N = 3 & width N = 3 by MATRIX_0:24;
A20: 1.(F_Real,3) * (<*uf*>@) = (N~) * N * (<*uf*>@) by A17,MATRIX_6:def 2
                             .= (N~) * (N * (<*uf*>@))
                                 by A9,A18,A19,MATRIX_3:33;
A21: N * (<*uf*>@) is Matrix of 3,1,F_Real by A12,MATRIX_0:20;
    N * (<*uf*>@) = <* <* 0 *>, <* 0 *>, <* 0 *> *> by LAPLACE:def 9,A3,A16;
    then (N~) * (N * (<*uf*>@)) = <* <* 0 *>, <* 0 *>, <* 0 *> *>
      by Th7,A21;
    then <*uf*>@ = <* <* 0 *>, <* 0 *>, <* 0 *> *>
      by A20,Th80,A5,EUCLID_8:50;
    hence thesis by A1,A2,Th81;
  end;
