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 Th6:
  A + (-A) = 0.(F,n)
proof
A1: Indices A= Indices (A + (-A)) 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 + (-A));
    hence (A + (-A))*(i,j)=A*(i,j)+ (-A)*(i,j) by A1,Def5
      .=A*(i,j)+ (-A*(i,j)) by A1,A3,Def4
      .=0.F by RLVECT_1:def 10
      .=(0.(F,n))*(i,j) by A2,A1,A3,Th1;
  end;
  hence thesis by MATRIX_0:27;
end;
