theorem Th5:
  A + 0.(F,n)= A
proof
A1: Indices A= Indices (A+0.(F,n)) by MATRIX_0:26;
A2: Indices A= Indices (0.(F,n)) by MATRIX_0:26;
  now
    let i,j;
    assume
A3: [i,j] in Indices (A+ 0.(F,n));
    hence (A+ 0.(F,n)) *(i,j)=A*(i,j) + 0.(F,n)*(i,j) by A1,Def5
      .=A*(i,j) +0.F by A1,A2,A3,Th1
      .=A*(i,j) by RLVECT_1:4;
  end;
  hence thesis by MATRIX_0:27;
end;
