reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  K for non empty doubleLoopStr,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D,
  F for add-associative right_zeroed
  right_complementable Abelian non empty doubleLoopStr;
reserve A,B for Matrix of n,K;
reserve A,A9,B,B9,C for Matrix of n,F;

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;
