| p | + | r | , or | ps + qr |
|---|---|---|---|---|
| q | s | qs |
and (m + n)a is
| ps + qr | × | b | , or | ( ps + qr ) b | |
|---|---|---|---|---|---|
| qs | c | qsc | |||
| or | psb + qrb | . | |||
| qsc | |||||
| But this (112) is | psb | + | qrb | , which is | pb | + | rb | , |
|---|---|---|---|---|---|---|---|---|
| qsc | qsc | qc | sc | |||||
| since | psb | = | pb | , and | qrb | = | rb | (103). |
| qsc | qc | qsc | sc | |||||
| But | pb | = | p | × | b | , and | rb | , | = | r | × | b | . |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| qc | q | c | sc | s | c |
Therefore (m + n)a, or
| ( | p | + | r | ) | b | = | p | × | b | + | r | × | b | . |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| q | s | c | q | c | s | c |
In a similar manner the same may be proved of any other formula.
The following examples may be useful:
| a | × | c | + | e | × | g | = | acfh + bdeg |
|---|---|---|---|---|---|---|---|---|
| b | d | f | h | |||||
| ———————— | ————— | |||||||
| a | × | e | + | c | × | g | aedh + bcfg | |
| b | f | d | h |
| 1 | = | b | |
|---|---|---|---|
| 1 | ab + 1 | ||
| a | + | — | |
| b |
| 1 | = | 1 | = | bc + 1 | ||
|---|---|---|---|---|---|---|
| ———————— | —————— | —————— | ||||
| 1 | c | abc + a + c | ||||
| a + | ———— | a + | ——— | |||
| 1 | bc + 1 | |||||
| b + | — | |||||
| c |
Thus,