reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,a1,a2 for Element of K,
  D for non empty set,
  d,d1,d2 for Element of D,
  M,M1,M2 for (Matrix of D),
  A,A1,A2,B1,B2 for (Matrix of K),
  f,g for FinSequence of NAT;
reserve F,F1,F2 for FinSequence_of_Matrix of D,
  G,G9,G1,G2 for FinSequence_of_Matrix of K;
reserve S,S1,S2 for FinSequence_of_Square-Matrix of D,
  R,R1,R2 for FinSequence_of_Square-Matrix of K;
reserve N for (Matrix of n,K),
  N1 for (Matrix of m,K);
reserve p,p1 for FinSequence of K;

theorem Th70:
  <*A1,B1*>(+)<*A2,B2*> = <*A1+A2,B1+B2*>
proof
A1: len <*A2*>=1 by FINSEQ_1:39;
  len <*A1*>=1 by FINSEQ_1:39;
  hence (<*A1,B1*>)(+)(<*A2,B2*>) = (<*A1*>(+)<*A2*>)^(<*B1*>(+)<*B2*>) by A1
,Th67
    .= (<*A1*>(+)<*A2*>)^(<*B1+B2*>) by Th69
    .= <*A1+A2,B1+B2*> by Th69;
end;
