let L be Lattice; :: thesis: for I being Ideal of L st L is lower-bounded holds
latt (L,I) is lower-bounded
let I be Ideal of L; :: thesis: ( L is lower-bounded implies latt (L,I) is lower-bounded )
set b9 = the Element of (latt (L,I));
reconsider b = the Element of (latt (L,I)) as Element of L by Th68;
given c being Element of L such that A1: for a being Element of L holds
( c "/\" a = c & a "/\" c = c ) ; :: according to LATTICES:def 13 :: thesis: latt (L,I) is lower-bounded
A2: H₁( latt (L,I)) = I by Th72;
c "/\" b = c by A1;
then reconsider c9 = c as Element of (latt (L,I)) by A2, Th22;
take c9 ; :: according to LATTICES:def 13 :: thesis: for b₁ being Element of the carrier of (latt (L,I)) holds
( c9 "/\" b₁ = c9 & b₁ "/\" c9 = c9 )
let a9 be Element of (latt (L,I)); :: thesis: ( c9 "/\" a9 = c9 & a9 "/\" c9 = c9 )
reconsider a = a9 as Element of L by Th68;
thus c9 "/\" a9 = c "/\" a by Th73
.= c9 by A1 ; :: thesis: a9 "/\" c9 = c9
hence a9 "/\" c9 = c9 ; :: thesis: verum