theorem
  for m,n being Nat
  for M being Matrix of m,F_Real
  for N being Matrix of m,n,F_Real st m > 0 holds
  M * N is Matrix of m,n,F_Real
  proof
    let m,n be Nat;
    let M be Matrix of m,F_Real;
    let N be Matrix of m,n,F_Real;
    assume
A1: m > 0;
    len N = m & width N = n by A1,MATRIX_0:23;
    then width M = len N by A1,MATRIX_0:23;
    then len(M * N) = len M & width(M * N) = width N by MATRIX_3:def 4;
    then len(M * N) = m & width(M * N) = n by A1,MATRIX_0:23;
    hence thesis by A1,MATRIX_0:20;
  end;
