Hence, The Isomorphism between Two-Well-Ordered Sets W1 and W2 is UNIQUE
Make a Plan: Understand the Problem
We need to Prove if Two-Well-Ordered Sets W1 and W2 are ISOMORPHIC, Then there is ONLY ONE ISOMORPHISM.
PROVE the Statement
1 -Assume there are Two Isomorphisms f and g between W1 and W2
2 - Define a Set:
A = f(x) ≠ g(x)
3 - If A is Empty, Then f(x) = g(x) for all x ∈ W1, and the Isomorphism is Unique.
4 - If A is Non-Empty, Then Since W1 is Well-Ordered, There Exist a Least Element a ∈ A
5 - Since f and g are Isomorphisms, f(a) and g(a) are the Least Elements in their respective images in W2.
6 - Since f(a) ≠ g(a), One of them, say f(a), is less than g(a).
7 - Let b = f^-1(g(a)). Then b > a because f is an Isomorphism and f(a) < g(a).
8 - Since b > a, We now have f(b) = g(b) because, a is the Least Element in A.
9 - However, f(b) = g(a), Which Contradicts the Fact that f and g are Isomorphisms
Hence, The Isomorphism between Two-Well-Ordered Sets W1 and W2 is UNIQUE
I hope this helps you!