let a, b, d be Element of dL; :: thesis: ( a * (b + d) = (a * b) + (a * d) & (b + d) * a = (b * a) + (d * a) )
A1: ( ( a = In 0 ,2 & b = In 0 ,2 & d = In 0 ,2 ) or ( a = In 0 ,2 & b = In 0 ,2 & d = In 1,2 ) or ( a = In 0 ,2 & b = In 1,2 & d = In 0 ,2 ) or ( a = In 0 ,2 & b = In 1,2 & d = In 1,2 ) or ( a = In 1,2 & b = In 0 ,2 & d = In 0 ,2 ) or ( a = In 1,2 & b = In 0 ,2 & d = In 1,2 ) or ( a = In 1,2 & b = In 1,2 & d = In 0 ,2 ) or ( a = In 1,2 & b = In 1,2 & d = In 1,2 ) ) by Lm2, CARD_1:88, TARSKI:def 2;
hence a * (b + d) = (a * b) + (a * d) by Lm6, Lm7, Lm8, Lm9, Lm13, Lm14, Lm15, Lm16; :: thesis: (b + d) * a = (b * a) + (d * a)
thus (b + d) * a = (b * a) + (d * a) by A1, Lm6, Lm7, Lm8, Lm9, Lm13, Lm14, Lm15, Lm16; :: thesis: verum