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 Th25:
  MD = LineVec2Mx bD iff Line(MD,1) = bD & len MD = 1
proof
  thus MD = LineVec2Mx bD implies Line(MD,1) = bD & len MD = 1
  proof
    1 in Seg 1;
    then
A1: Line(LineVec2Mx bD,1)=(LineVec2Mx bD).1 by MATRIX_0:52;
    assume MD = LineVec2Mx bD;
    hence thesis by A1,FINSEQ_1:40;
  end;
  assume that
A2: Line(MD,1) = bD and
A3: len MD = 1;
  reconsider md=MD as Matrix of 1,width MD,D by A3,MATRIX_0:51;
  1 in Seg 1;
  then md.1=bD by A2,MATRIX_0:52;
  hence thesis by A3,FINSEQ_1:40;
end;
