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

theorem Th11:
  for D being set, e being FinSequence of D* holds (ex n st for i
  st i in dom e holds len(e.i) = n) iff e is Matrix of D
proof
  let D be set, e be FinSequence of D*;
  hereby
    given n such that
A1: for i st i in dom e holds len(e.i) = n;
    for i st i in dom e holds ex p being FinSequence of D st e.i = p & len
    p = n
    proof
      let i;
      assume i in dom e;
      then len (e.i) = n by A1;
      hence thesis;
    end;
    hence e is Matrix of D by Th10;
  end;
  assume e is Matrix of D;
  then consider n such that
A2: for i st i in dom e holds ex p being FinSequence of D st e.i = p &
  len p = n by Th10;
  for i st i in dom e holds len(e.i) = n
  proof
    let i;
    assume i in dom e;
    then ex p being FinSequence of D st e.i = p & len p = n by A2;
    hence thesis;
  end;
  hence thesis;
end;
