# Equality of Ordered Triples/Proof 2

## Example of Equality of Ordered Tuples

Let:

$\tuple {a_1, a_2, a_3}$ and $\tuple {b_1, b_2, b_3}$

Then:

$\tuple {a_1, a_2, a_3} = \tuple {b_1, b_2, b_3}$
$\forall i \in \set {1, 2, 3}: a_i = b_i$

## Proof

 $\ds A$ $=$ $\ds B$ $\ds \leadstoandfrom \ \$ $\ds \tuple {a_1, a_2, a_3}$ $=$ $\ds \tuple {b_1, b_2, b_3}$ Definition of $A$ and $B$ $\ds \leadstoandfrom \ \$ $\ds \tuple {a_1, \tuple {a_2, a_3} }$ $=$ $\ds \tuple {b_1, \tuple {b_2, b_3} }$ Definition of Ordered Triple $\ds \leadstoandfrom \ \$ $\ds a_1$ $=$ $\ds b_1$ Equality of Ordered Pairs $\, \ds \land \,$ $\ds \tuple {a_2, a_3}$ $=$ $\ds \tuple {b_2, b_3}$ $\ds \leadstoandfrom \ \$ $\ds a_1$ $=$ $\ds b_1$ Equality of Ordered Pairs $\, \ds \land \,$ $\ds a_2$ $=$ $\ds b_2$ $\, \ds \land \,$ $\ds a_3$ $=$ $\ds b_3$

$\blacksquare$