let L1, L2 be Lattice; :: thesis: ( ( L1 is lower-bounded & L2 is lower-bounded ) iff [:L1,L2:] is lower-bounded )
thus ( L1 is lower-bounded & L2 is lower-bounded implies [:L1,L2:] is lower-bounded ) :: thesis: ( [:L1,L2:] is lower-bounded implies ( L1 is lower-bounded & L2 is lower-bounded ) )
proof
given p1 being Element of L1 such that A1: for q1 being Element of L1 holds
( p1 "/\" q1 = p1 & q1 "/\" p1 = p1 ) ; :: according to LATTICES:def 13 :: thesis: ( not L2 is lower-bounded or [:L1,L2:] is lower-bounded )
given p2 being Element of L2 such that A2: for q2 being Element of L2 holds
( p2 "/\" q2 = p2 & q2 "/\" p2 = p2 ) ; :: according to LATTICES:def 13 :: thesis: [:L1,L2:] is lower-bounded
take a = [p1,p2]; :: according to LATTICES:def 13 :: thesis: for b1 being Element of the carrier of [:L1,L2:] holds
( a "/\" b1 = a & b1 "/\" a = a )
let b be Element of [:L1,L2:]; :: thesis: ( a "/\" b = a & b "/\" a = a )
consider q1 being Element of L1, q2 being Element of L2 such that
A3: b = [q1,q2] by DOMAIN_1:1;
thus a "/\" b = [(p1 "/\" q1),(p2 "/\" q2)] by A3, Th21
.= [p1,(p2 "/\" q2)] by A1
.= a by A2 ; :: thesis: b "/\" a = a
hence b "/\" a = a ; :: thesis: verum
end;
given a being Element of [:L1,L2:] such that A4: for b being Element of [:L1,L2:] holds
( a "/\" b = a & b "/\" a = a ) ; :: according to LATTICES:def 13 :: thesis: ( L1 is lower-bounded & L2 is lower-bounded )
consider p1 being Element of L1, p2 being Element of L2 such that
A5: a = [p1,p2] by DOMAIN_1:1;
thus L1 is lower-bounded :: thesis: L2 is lower-bounded
proof
set q2 = the Element of L2;
take p1 ; :: according to LATTICES:def 13 :: thesis: for b1 being Element of the carrier of L1 holds
( p1 "/\" b1 = p1 & b1 "/\" p1 = p1 )
let q1 be Element of L1; :: thesis: ( p1 "/\" q1 = p1 & q1 "/\" p1 = p1 )
a = a "/\" [q1, the Element of L2] by A4
.= [(p1 "/\" q1),(p2 "/\" the Element of L2)] by A5, Th21 ;
hence ( p1 "/\" q1 = p1 & q1 "/\" p1 = p1 ) by A5, XTUPLE_0:1; :: thesis: verum
end;
set q1 = the Element of L1;
take p2 ; :: according to LATTICES:def 13 :: thesis: for b1 being Element of the carrier of L2 holds
( p2 "/\" b1 = p2 & b1 "/\" p2 = p2 )
let q2 be Element of L2; :: thesis: ( p2 "/\" q2 = p2 & q2 "/\" p2 = p2 )
a = a "/\" [ the Element of L1,q2] by A4
.= [(p1 "/\" the Element of L1),(p2 "/\" q2)] by A5, Th21 ;
hence ( p2 "/\" q2 = p2 & q2 "/\" p2 = p2 ) by A5, XTUPLE_0:1; :: thesis: verum