Hypothesis: Equal numbers a and b of the same magnitude need not be equal i.e 1+1 is not necessarily equal to 3-1, even though numerically they are both equal to 2. Method of proof: By contradiction Proof A: I have 3 million dollars, but I lost 1 million dollars, so I still have 2 million dollars. I might go jump down. If I have 1 million dollar and I made another million, I now have 2 million dollars, instead I jump for joy. The former makes me jump down, the latter makes me jump up, possibly with fist pumping and with occasional shouts of joy. Thus they are clearly different, even though it's the same. Proof B: I have 3 bad debts (all of equal amount), and I tried all ways to get rid of 1, so now I have 2 bad debts. I'm overjoyed. ......