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 Th3:
  A + B = B + A
proof
A1: Indices A = Indices (A + B) by MATRIX_0:26;
A2: Indices A= Indices B by MATRIX_0:26;
  now
    let i,j;
    assume
A3: [i,j] in Indices (A + B);
    hence (A + B)*(i,j)=A*(i,j) + B*(i,j) by A1,Def5
      .=(B + A)*(i,j) by A2,A1,A3,Def5;
  end;
  hence thesis by MATRIX_0:27;
end;
