let L be complete Lattice; :: thesis: for L9 being SubLattice of L st L9 is /\-inheriting holds
for A9 being Subset of L9 holds "/\" A9,L = "/\" A9,L9
let L9 be SubLattice of L; :: thesis: ( L9 is /\-inheriting implies for A9 being Subset of L9 holds "/\" A9,L = "/\" A9,L9 )
assume A1: L9 is /\-inheriting ; :: thesis: for A9 being Subset of L9 holds "/\" A9,L = "/\" A9,L9
then reconsider L91 = L9 as complete SubLattice of L ;
let A9 be Subset of L9; :: thesis: "/\" A9,L = "/\" A9,L9
set a = "/\" A9,L;
reconsider a9 = "/\" A9,L as Element of L91 by A1, Def2;

A2: now

let c9 be Element of L91; :: thesis: ( c9 is_less_than A9 implies c9 [= a9 )
the carrier of L91 c= the carrier of L by NAT_LAT:def 16;
then reconsider c = c9 as Element of L by TARSKI:def 3;
assume c9 is_less_than A9 ; :: thesis: c9 [= a9
then c is_less_than A9 by Th13;
then A3: c [= "/\" A9,L by LATTICE3:34;
c9 "/\" a9 = c "/\" ("/\" A9,L) by Th12
.= c9 by A3, LATTICES:21 ;
hence c9 [= a9 by LATTICES:21; :: thesis: verum

end;

"/\" A9,L is_less_than A9 by LATTICE3:34;
then a9 is_less_than A9 by Th13;
hence "/\" A9,L = "/\" A9,L9 by A2, LATTICE3:34; :: thesis: verum