let V, C be set ; :: thesis: for a, b being Element of (SubstLatt V,C) holds a "/\" (a "\/" b) = a
let a, b be Element of (SubstLatt V,C); :: thesis: a "/\" (a "\/" b) = a
reconsider a' = a, b' = b as Element of SubstitutionSet V,C by Def4;
thus a "/\" (a "\/" b) = the L_join of (SubstLatt V,C) . (the L_meet of (SubstLatt V,C) . a',a'),(the L_meet of (SubstLatt V,C) . a',b') by Lm12
.= the L_join of (SubstLatt V,C) . (mi (a' ^ a')),(the L_meet of (SubstLatt V,C) . a',b') by Def4
.= the L_join of (SubstLatt V,C) . (mi a'),(the L_meet of (SubstLatt V,C) . a',b') by Th24
.= a "\/" (a "/\" b) by Th11
.= (a "/\" b) "\/" a by Lm6
.= (b "/\" a) "\/" a by Lm10
.= a by Lm9 ; :: thesis: verum