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

theorem Th18:
  for e1 being FinSequence of D st len e1 = m holds n |-> e1 is Matrix of n,m,D
proof
  let e1 be FinSequence of D such that
A1: len e1 = m;
  reconsider e = n |-> e1 as FinSequence of D* by Th8;
A2: len e = n by CARD_1:def 7;
A3: 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,FUNCOP_1:7;
  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 A3;
  end;
  hence thesis by A2,MATRIX_0:def 2;
end;
