reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;

theorem Th1:
  for A be Matrix of n,m,D holds (n=0 implies m=0) iff len A = n & width A = m
proof
  let A be Matrix of n,m,D;
  thus (n=0 implies m=0) implies len A = n & width A = m
  proof
    assume
A1: n=0 implies m=0;
    per cases;
    suppose
A2:   n=0;
      then len A=0 by MATRIX_0:def 2;
      hence thesis by A1,A2,MATRIX_0:def 3;
    end;
    suppose
A3:   n>0;
      len A=n by MATRIX_0:def 2;
      hence thesis by A3,MATRIX_0:20;
    end;
  end;
  thus thesis by MATRIX_0:def 3;
end;
