
theorem LMEQ:
  for D, E being non empty set, n, m being Nat, M being Matrix of n,m,D
  st for i, j being Nat st [i, j] in Indices M
  holds M*(i,j) in E
  holds M is Matrix of n,m,E
  proof
    let D, E be non empty set, n, m be Nat, M be Matrix of n,m,D;
    assume for i, j being Nat st [i, j] in Indices M
    holds M*(i,j) in E;
    then reconsider G = M as Matrix of E by ZMATRLIN:5;
    per cases;
    suppose not n > 0; then
      X1: n = 0;
      then G = {} by MATRIX_0:22;
      hence thesis by X1,MATRIX_0:13;
    end;
    suppose X1: n > 0;
      then len G = n & width G = m by MATRIX_0:23;
      hence thesis by X1,MATRIX_0:20;
    end;
  end;
