reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);
reserve D for non empty set,
  bD for FinSequence of D,
  b,f,g for FinSequence of K,
  MD for Matrix of D;

theorem Th26:
  ( len MD <> 0 or len bD <> 0 ) implies ( MD = ColVec2Mx bD iff
  Col(MD,1) = bD & width MD = 1 )
proof
  assume
A1: len MD <> 0 or len bD <> 0;
  thus MD = ColVec2Mx bD implies Col(MD,1) = bD & width MD = 1
  proof
    len (LineVec2Mx bD)=1 by Th25;
    then
A2: dom (LineVec2Mx bD)=Seg 1 by FINSEQ_1:def 3;
    assume
A3: MD = ColVec2Mx bD;
    1 in Seg 1;
    hence Col(MD,1) = Line(LineVec2Mx bD,1) by A3,A2,MATRIX_0:58
      .= bD by Th25;
    len MD=len bD by A3,MATRIX_0:def 2;
    hence thesis by A1,A3,MATRIX_0:23;
  end;
  assume that
A4: Col(MD,1) = bD and
A5: width MD = 1;
A6: len MD>0 by A1,A4,MATRIX_0:def 8;
A7: len (MD@)=1 by A5,MATRIX_0:def 6;
  1 in Seg 1;
  then Line(MD@,1)=bD by A4,A5,MATRIX_0:59;
  then (LineVec2Mx bD)@ = (MD@)@ by A7,Th25
    .= MD by A5,A6,MATRIX_0:57;
  hence thesis;
end;
