let L1, L2 be Lattice; :: thesis: ( L1 is 0_Lattice & L2 is 0_Lattice implies Bottom [:L1,L2:] = [(Bottom L1),(Bottom L2)] )
assume that
A1: L1 is 0_Lattice and
A2: L2 is 0_Lattice ; :: thesis: Bottom [:L1,L2:] = [(Bottom L1),(Bottom L2)]
A3: now :: thesis: for a being Element of [:L1,L2:] holds
( [(Bottom L1),(Bottom L2)] "/\" a = [(Bottom L1),(Bottom L2)] & a "/\" [(Bottom L1),(Bottom L2)] = [(Bottom L1),(Bottom L2)] )
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
A4: a = [p1,p2] by DOMAIN_1:1;
thus [(Bottom L1),(Bottom L2)] "/\" a = [((Bottom L1) "/\" p1),((Bottom L2) "/\" p2)] by A4, Th21
.= [(Bottom L1),((Bottom L2) "/\" p2)] by A1
.= [(Bottom L1),(Bottom L2)] by A2 ; :: 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, A2, Th39;
hence Bottom [:L1,L2:] = [(Bottom L1),(Bottom L2)] by A3, LATTICES:def 16; :: thesis: verum