If I have a partial order, is the following conclusion valid? $$a \geq c$$ $$b \geq c$$
Then, $$a = b $$
Does the result change if the partial order were to become a total order? Any and all explanations would be helpful. Cheers.
Follow-up: This seems quite obvious but I'm curious nonetheless. First, call S a finite subset of the naturals. Now define a and b with the same definition:
$$a, b \geq n$$ for all n in S. Does it follow that a = b by definition? From a method standpoint, is it enough to draw an equality between two elements by showing that those elements are defined in the same way?