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 Lm3, Lm4, CARD_1:50, TARSKI:def 2;
hence a * (b + d) = (a * b) + (a * d) by Lm8, Lm9, Lm10, Lm11, Lm15, Lm16, Lm17, Lm18; :: thesis: (b + d) * a = (b * a) + (d * a)
thus (b + d) * a = (b * a) + (d * a) by A1, Lm8, Lm9, Lm10, Lm11, Lm15, Lm16, Lm17, Lm18; :: thesis: verum