prove that the 2n+1 construction gives a Steiner triple system of order 2n+1

Q1 : prove that the 2n+1 construction gives a Steiner triple system of order 2n+1 ? 

“The book design theory second edition

Steps, prove that (S,T) is a Steiner triple system

Type 1 triples is 2n+1 and the defining type 2triples is (2n+1)(2n)/2 choices for

I and j

Therefore ǀ T ǀ = (2n+1) + 3(2n+1)(2n)/2

                          = (2n+1)(3n+1)

                          =v(v-1)/6

Therefore T contains the right number of triples, therefore each pair of distinct symbols in S occurs together in at least one triple of T