reserve x,y,z,x1,x2,y1,y2,z1,z2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Ring;

theorem Th1:
  for i,j st [i,j] in Indices (0.(K,n,m)) holds (0.(K,n,m))*(i,j)= 0.K
proof
  let i,j;
  assume
A1: [i,j] in Indices (0.(K,n,m));
A2: Indices (0.(K,n,m))= [:Seg n,Seg width (0.(K,n,m)):] by MATRIX_0:25;
  per cases;
  suppose
A3: n > 0;
    reconsider m1=m as Element of NAT by ORDINAL1:def 12;
A4: [i,j] in [:Seg n,Seg m:] by A1,A3,MATRIX_0:23;
    then j in Seg m by ZFMISC_1:87;
    then
A5: (m1 |-> 0.K).j= 0.K by FUNCOP_1:7;
    i in Seg n by A4,ZFMISC_1:87;
    then (0.(K,n,m)).i= m |-> 0.K by FUNCOP_1:7;
    hence thesis by A1,A5,MATRIX_0:def 5;
  end;
  suppose
    n=0;
    then Seg n = {};
    hence thesis by A1,A2,ZFMISC_1:90;
  end;
end;
