We need to prove x+yz=(x+y)(x+z) for all permutations of Boolean variables x, y, z.
Multiplication is understood to be logical AND, and addition to be logical OR (so 1+1=1, not 2).
We can do this by setting up a table to cover all possibilities. There are 8 variants of x, y, z.
x |
y |
z |
x+yz |
x+y |
x+z |
(x+y)(x+z) |
0 |
0 |
0 |
0+0=0 |
0 |
0 |
0 |
0 |
0 |
1 |
0+0=0 |
0 |
1 |
0 |
0 |
1 |
0 |
0+0=0 |
1 |
0 |
0 |
0 |
1 |
1 |
0+1=1 |
1 |
1 |
1 |
1 |
0 |
0 |
1+0=1 |
1 |
1 |
1 |
1 |
0 |
1 |
1+0=1 |
1 |
1 |
1 |
1 |
1 |
0 |
1+0=1 |
1 |
1 |
1 |
1 |
1 |
1 |
1+1=1 |
1 |
1 |
1 |
From the table it's clear that, since the results in columns 4 and 7 are the same, that the Distributive Law applies.