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 )
consider b9 being Element of (latt L,I);
reconsider b = b9 as Element of L by Th69;
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 Th73;
c "/\" b = c by A1;
then reconsider c9 = c as Element of (latt L,I) by A2, Th23;
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 Th69;
thus c9 "/\" a9 = c "/\" a by Th74
.= c9 by A1 ; :: thesis: a9 "/\" c9 = c9
hence a9 "/\" c9 = c9 ; :: thesis: verum