let X be Tolerance_Space; :: thesis: for a, b, c being Element of holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
let a, b, c be Element of ; :: thesis: a "\/" (b "\/" c) = (a "\/" b) "\/" c
reconsider a1 = a, b1 = b, c1 = c as Element of RoughSets X by Def8;
reconsider a' = a1, b' = b1, c' = c1 as RoughSet of X by DefRSX;
set G = RSLattice X;
XX: b' _\/_ c' is Element of RoughSets X by DefRSX;
XY: a' _\/_ b' is Element of RoughSets X by DefRSX;
a "\/" (b "\/" c) = the L_join of (RSLattice X) . a,(b' _\/_ c') by Def8
.= a' _\/_ (b' _\/_ c') by Def8, XX
.= (a' _\/_ b') _\/_ c' by Th7
.= the L_join of (RSLattice X) . (a' _\/_ b'),c1 by Def8, XY
.= (a "\/" b) "\/" c by Def8 ;
hence a "\/" (b "\/" c) = (a "\/" b) "\/" c ; :: thesis: verum