Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the identity operator, which does nothing): T ⁡ ⁡ ( 0 ) = 1 T ⁡ ⁡ ( a ) T ⁡ ⁡ ( d a ) | x ⟩ ⟩ = T ⁡ ⁡ ( a ) | x + d a ⟩ ⟩ = | x + a + d a ⟩ ⟩ = T ⁡ ⁡ ( a + d a ) | x ⟩ ⟩ ⇒ ⇒ T ⁡ ⁡ ( a ) T ⁡ ⁡ ( d a ) = T ⁡ ⁡ ( a + d a )