let L1, L2 be Lattice; :: thesis: ( L1 is 0_Lattice & L2 is 0_Lattice implies Bottom [:L1,L2:] = [(Bottom L1),(Bottom L2)] )
assume A1: ( L1 is 0_Lattice & L2 is 0_Lattice ) ; :: thesis: Bottom [:L1,L2:] = [(Bottom L1),(Bottom L2)]

A2: now

let a be Element of [:L1,L2:]; :: thesis: ( [(Bottom L1),(Bottom L2)] "/\" a = [(Bottom L1),(Bottom L2)] & a "/\" [(Bottom L1),(Bottom L2)] = [(Bottom L1),(Bottom L2)] )
consider p1 being Element of L1, p2 being Element of L2 such that
A3: a = [p1,p2] by DOMAIN_1:9;
thus [(Bottom L1),(Bottom L2)] "/\" a = [((Bottom L1) "/\" p1),((Bottom L2) "/\" p2)] by A3, Th22
.= [(Bottom L1),((Bottom L2) "/\" p2)] by A1, LATTICES:40
.= [(Bottom L1),(Bottom L2)] by A1, LATTICES:40 ; :: thesis: a "/\" [(Bottom L1),(Bottom L2)] = [(Bottom L1),(Bottom L2)]
hence a "/\" [(Bottom L1),(Bottom L2)] = [(Bottom L1),(Bottom L2)] ; :: thesis: verum

end;

[:L1,L2:] is lower-bounded by A1, Th40;
hence Bottom [:L1,L2:] = [(Bottom L1),(Bottom L2)] by A2, LATTICES:def 16; :: thesis: verum