reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;

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;
