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;
