theorem Th32:
  for N being Matrix of 3,REAL
  for uf being FinSequence of REAL
  for u being Element of TOP-REAL 3
  st N is invertible & u = uf & u is non zero
  holds N * uf <> 0.TOP-REAL 3
  proof
    let N be Matrix of 3,REAL;
    let uf be FinSequence of REAL;
    let u be Element of TOP-REAL 3;
    assume
A1: N is invertible & u = uf & u is non zero;
    then dom uf = Seg 3 by FINSEQ_1:89; then
A2: len uf = 3 by FINSEQ_1:def 3;
    assume
A3: N * uf = 0.TOP-REAL 3;
A4: (Inv N) * N = 1_Rmatrix(3) & (Inv N) * N = 1_Rmatrix(3)
      by A1,MATRIXR2:def 6;
    width N = 3 & len N = 3 & width Inv(N) = 3 by MATRIX_0:24;
    then Inv(N) * (N * uf) = (Inv(N) * N) * uf by A2,MATRIXR2:59
                          .= uf by A4,A2,MATRIXR2:86;
    hence contradiction by A3,A1,Th31;
  end;
