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);

theorem Th7:
  for M being Matrix of 3,F_Real
  for N being Matrix of 3,1,F_Real st N = <* <* 0 *>, <* 0 *>, <* 0 *> *> holds
  M * N = <* <* 0 *>, <* 0 *>, <* 0 *> *>
  proof
    let M be Matrix of 3,F_Real;
    let N be Matrix of 3,1,F_Real;
    assume
A1: N = <* <* 0 *>, <* 0 *>, <* 0 *> *>;
A2: len M = 3 & width M = 3 by MATRIX_0:23;
A3: len N = 3 & width N = 1 by MATRIX_0:23;
    width M = len N by A3,MATRIX_0:23; then
A4A: len(M * N) = len M & width(M * N) = width N by MATRIX_3:def 4; then
A4: len(M * N) = 3 & width(M * N) = 1 by MATRIX_0:23; then
A5: M * N is Matrix of 3,1,F_Real by MATRIX_0:20;
    now
      thus len Line(M,1) = width M by MATRIX_0:def 7
                        .= 3 by MATRIX_0:23;
      1 in Seg width M & 2 in Seg width M &
       3 in Seg width M by A2,FINSEQ_1:1;
      hence Line(M,1).1 = M*(1,1) &
        Line(M,1).2 = M*(1,2) & Line(M,1).3 = M*(1,3)  by MATRIX_0:def 7;
    end;
    then
A6: Line(M,1) = <* M*(1,1),M*(1,2),M*(1,3) *> by FINSEQ_1:45;
    reconsider ze = 0 as Element of F_Real;
    <* M*(1,1),M*(1,2),M*(1,3) *> "*" <* ze,ze,ze *> =
    M*(1,1) * ze + M*(1,2) * ze + M*(1,3) * ze by Th6; then
A7: Line(M,1) "*" Col(N,1) = 0 by A1,Th4,A6;
    now
A8:   1 in Seg 3 by FINSEQ_1:1;
      len Line(M * N,1) = width (M * N) by MATRIX_0:def 7
                       .= 1 by A4A,MATRIX_0:23; then
A9:   Line(M * N,1) = <* (Line(M * N,1)).1 *> by FINSEQ_1:40;
      2 in Seg 3 by FINSEQ_1:1; then
A10:  (M * N).2 = Line(M * N,2) by A5,MATRIX_0:52;
A11:  len Line(M * N,2) = width (M * N) by MATRIX_0:def 7
                          .= 1 by A4A,MATRIX_0:23;
      3 in Seg 3 by FINSEQ_1:1; then
A12:  (M * N).3 = Line(M * N,3) by A5,MATRIX_0:52;
A13:  len Line(M * N,3) = width (M * N) by MATRIX_0:def 7
                          .= 1 by A4A,MATRIX_0:23;
      (Line(M * N,1)).1 = 0
      proof
A14:    [1,1] in Indices(M * N) by A5,MATRIX_0:23,Th2;
        1 in Seg width (M * N) by A4,FINSEQ_1:1; then
        (Line(M * N,1)).1 = (M * N)*(1,1) by MATRIX_0:def 7
                         .= 0 by A7,A14,A2,A3,MATRIX_3:def 4;
        hence thesis;
      end;
      hence (M * N).1 = <* 0 *> by A9,A8,A5,MATRIX_0:52;
      now
        thus len Line(M,2) = width M by MATRIX_0:def 7
                          .= 3 by MATRIX_0:23;
        1 in Seg width M & 2 in Seg width M &
          3 in Seg width M by A2,FINSEQ_1:1;
        hence Line(M,2).1 = M*(2,1) & Line(M,2).2 = M*(2,2) &
          Line(M,2).3 = M*(2,3) by MATRIX_0:def 7;
      end;
      then
A15:  Line(M,2) = <* M*(2,1),M*(2,2),M*(2,3) *> by FINSEQ_1:45;
      reconsider ze = 0 as Element of F_Real;
      <* M*(2,1),M*(2,2),M*(2,3) *> "*" <* ze,ze,ze *>
        = M*(2,1) * ze + M*(2,2) * ze + M*(2,3) * ze by Th6; then
A16:  Line(M,2) "*" Col(N,1) = 0 by A1,Th4,A15;
      (Line(M * N,2)).1 = 0
      proof
A17:    [2,1] in Indices(M * N) by A5,MATRIX_0:23,Th2;
        1 in Seg width (M * N) by A4,FINSEQ_1:1;
        then (Line(M * N,2)).1 = (M * N)*(2,1) by MATRIX_0:def 7
                              .= 0 by A16,A17,A2,A3,MATRIX_3:def 4;
        hence thesis;
      end;
      hence (M * N).2 = <* 0 *> by FINSEQ_1:40,A10,A11;
      now
        thus len Line(M,3) = width M by MATRIX_0:def 7
                          .= 3 by MATRIX_0:23;
        1 in Seg width M & 2 in Seg width M &
          3 in Seg width M by A2,FINSEQ_1:1;
        hence Line(M,3).1 = M*(3,1) & Line(M,3).2 = M*(3,2) &
          Line(M,3).3 = M*(3,3)  by MATRIX_0:def 7;
      end;
      then
A18:  Line(M,3) = <* M*(3,1),M*(3,2),M*(3,3) *> by FINSEQ_1:45;
      reconsider ze = 0 as Element of F_Real;
      <* M*(3,1),M*(3,2),M*(3,3) *> "*" <* ze,ze,ze *> =
        M*(3,1) * ze + M*(3,2) * ze + M*(3,3) * ze by Th6; then
A19:  Line(M,3) "*" Col(N,1) = 0 by A1,Th4,A18;
      (Line(M * N,3)).1 = 0
      proof
A20:    [3,1] in Indices(M * N) by A5,MATRIX_0:23,Th2;
        1 in Seg width (M * N) by A4,FINSEQ_1:1; then
        (Line(M * N,3)).1 = (M * N)*(3,1) by MATRIX_0:def 7
                         .= 0 by A20,A2,A3,MATRIX_3:def 4,A19;
        hence thesis;
      end;
      hence (M * N).3 = <* 0 *> by A13,FINSEQ_1:40,A12;
    end;
    hence thesis by A4A,MATRIX_0:23,FINSEQ_1:45;
  end;
