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;
reserve i,j,n for Nat,
  K for Field,
  a,b for Element of K;

theorem
  ((n,n)-->a) + ((n,n)-->b) = (n,n)-->(a+b)
proof
A1: Indices ((n,n)-->a) = Indices ((n,n)-->(a+b)) by MATRIX_0:26;
A2: Indices ((n,n)-->a) = Indices ((n,n)-->b) by MATRIX_0:26;
  now
    let i,j;
    assume
A3: [i,j] in Indices ((n,n)-->(a+b));
    then ((n,n)-->a)*(i,j)=a by A1,MATRIX_0:46;
    then ((n,n)-->a)*(i,j) +((n,n)-->b)*(i,j)=a+b by A2,A1,A3,MATRIX_0:46;
    hence ((n,n)-->a)*(i,j) +((n,n)-->b)*(i,j)=((n,n)-->(a+b))*(i,j)
                by A3,MATRIX_0:46;
  end;
  hence thesis by A1,Def5;
end;
