reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th19:
  for e1,e2 being FinSequence of D st len e1 = m & len e2 = m
holds ex M being Matrix of n,m,D st for i be Nat st i in Seg n holds (i in Seg
  k implies M.i = e1) & (not i in Seg k implies M.i = e2)
proof
  let e1,e2 be FinSequence of D such that
A1: len e1 = m & len e2 = m;
  consider e being FinSequence of D* such that
A2: len e = n and
A3: for i be Nat st i in Seg n holds (i in Seg k implies e.i = e1) & (
  not i in Seg k implies e.i = e2) by Th9;
A4: for i st i in dom e holds len(e.i) = m
  proof
    let i;
    assume i in dom e;
    then i in Seg n by A2,FINSEQ_1:def 3;
    hence thesis by A1,A3;
  end;
  then reconsider e as Matrix of D by Th11;
  for p being FinSequence of D st p in rng e holds len p = m
  proof
    let p be FinSequence of D;
    assume p in rng e;
    then ex i be object st i in dom e & p = e.i by FUNCT_1:def 3;
    hence thesis by A4;
  end;
  then e is Matrix of n,m,D by A2,MATRIX_0:def 2;
  hence thesis by A3;
end;
