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 Th15:
  for i,j st [i,j] in Indices (1.(K,n)) holds
    (i=j implies (Line((1.(K,n)),i)).j = 1.K) &
    (i<>j implies (Line((1.(K,n)),i)).j = 0.K)
proof
  let i,j;
  set B=1.(K,n);
  assume
A1: [i,j] in Indices (1.(K,n));
  then j in Seg width B by ZFMISC_1:87;
  then
A2: (Line(B,i)).j = B*(i,j) by MATRIX_0:def 7;
  hence i=j implies (Line((1.(K,n)),i)).j=1.K by A1,MATRIX_1:def 3;
  thus thesis by A1,A2,MATRIX_1:def 3;
end;
