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;

theorem Th1:
  [i,j] in Indices (0.(K,n)) implies (0.(K,n))*(i,j)= 0.K
proof
  reconsider n1=n as Element of NAT by ORDINAL1:def 12;
  set M = 0.(K,n);
  assume
A1: [i,j] in Indices M;
  then
A2: [i,j] in [:Seg n,Seg n:] by MATRIX_0:24;
  then j in Seg n by ZFMISC_1:87;
  then
A3: (n1 |-> 0.K).j= 0.K by FUNCOP_1:7;
  i in Seg n by A2,ZFMISC_1:87;
  then M.i= n1 |-> 0.K by FUNCOP_1:7;
  hence thesis by A1,A3,MATRIX_0:def 5;
end;
