reserve i,j for Nat;

theorem
  for x being FinSequence of REAL, M being Matrix of REAL holds M=
  LineVec2Mx x iff Line(M,1)=x & len M=1
proof
  let x be FinSequence of REAL, M be Matrix of REAL;
  thus M=LineVec2Mx x implies Line(M,1)=x & len M=1
  proof
    assume
A1: M=LineVec2Mx x;
    then
A2: for j st j in dom x holds M*(1,j) = x.j by Def10;
A3: width M=len x by A1,Def10;
    then dom x = Seg width M by FINSEQ_1:def 3;
    hence thesis by A1,A3,A2,Def10,MATRIX_0:def 7;
  end;
  assume that
A4: Line(M,1)=x and
A5: len M=1;
A6: for j st j in Seg width M holds x.j = M*(1,j) by A4,MATRIX_0:def 7;
A7: len x = width M by A4,MATRIX_0:def 7;
  then Seg width M = dom x by FINSEQ_1:def 3;
  hence thesis by A5,A7,A6,Def10;
end;
