let A be set ; :: thesis: for a, b, c being Element of (NormForm A) holds a "/\" (b "/\" c) = (a "/\" b) "/\" c
set G = NormForm A;
let a, b, c be Element of (NormForm A); :: thesis: a "/\" (b "/\" c) = (a "/\" b) "/\" c
reconsider a9 = a, b9 = b, c9 = c as Element of Normal_forms_on A by Def12;
a "/\" (b "/\" c) = the L_meet of (NormForm A) . (a,(mi (b9 ^ c9))) by Def12
.= mi (a9 ^ (mi (b9 ^ c9))) by Def12
.= mi (a9 ^ (b9 ^ c9)) by Th51
.= mi ((a9 ^ b9) ^ c9) by Th52
.= mi ((mi (a9 ^ b9)) ^ c9) by Th50
.= the L_meet of (NormForm A) . ((mi (a9 ^ b9)),c9) by Def12
.= (a "/\" b) "/\" c by Def12 ;
hence a "/\" (b "/\" c) = (a "/\" b) "/\" c ; :: thesis: verum