0 cdot a = 0

Proof: proof here

A1:  a + b = b + a

A2:  left( a + b right) + c = a + left( b + c right)

A3:  a + 0 = a

A4: a + left( -a right) = 0

M1:  ab = ba

M2:  a left( bc right) = left( ab right) c

M3:  a cdot 1 = a

M4:  a cdot a^{-1} = 1

D1:  a left( b+ c right)= ab + ac

Theorems Proved on the next page:

1. 0 cdot a = 0
2. If a + b = a then b = 0
3. – left( -a right) = a
4. If a + b = 0 then a = -b
5. – left( a + b right) = left( -a right) + left( -b right)
6. left( -a right) left( -b right) = ab