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 Th68:
  <*A*>(+)G = <*A + G.1*>
proof
  dom (<*A*>(+) G)=dom <*A*> by Def10;
  then
A1: len (<*A*>(+) G) = len <*A*> by FINSEQ_3:29
    .= 1 by FINSEQ_1:39;
  then dom (<*A*>(+) G)={1} by FINSEQ_1:2,def 3;
  then 1 in dom (<*A*>(+) G) by TARSKI:def 1;
  then (<*A*>(+) G).1 = (<*A*>.1)+G.1 by Def10
    .= A+G.1;
  hence thesis by A1,FINSEQ_1:40;
end;
