**Theorem**:

- and implies that
- and implies that

**Proof**

*Part 1*

- Let
- that “ or ” is true
- “if x is in A, then x is in A or B”
- .

*Part 2*

- Let
- So “x in A and B implies that x is in A”
- .

*Part 3*

- Assume that and .
- Let .
- Then or .
- , because of the transitive property.
- , also by transitivity.
- So, .
- .

*Part 4*

- Assume and .
- Let .
- Since , .
- Since , .
- So, .
- .

*Now go lift something heavy,*

Nick Horton