
theorem u2:
for R being non degenerated Ring, n,i,j being Nat
st 1 <= i & i <= n & 1 <= j & j <= n
holds i_th_unit_vector(n,R) = j_th_unit_vector(n,R) iff i = j
proof
let R be non degenerated Ring, n,i,j be Nat;
assume AS: 1 <= i & i <= n & 1 <= j & j <= n;
now assume A: i_th_unit_vector(n,R) = j_th_unit_vector(n,R);
  now assume i <> j; then
    0.R = j_th_unit_vector(n,R).j by A,AS,u1;
    hence contradiction by AS,u1;
    end;
  hence i = j;
  end;
hence thesis;
end;
