let L be Lattice; :: thesis: for p, q, r being Element of L holds
( p is_less_than {q,r} iff p [= q "/\" r )
let p, q, r be Element of L; :: thesis: ( p is_less_than {q,r} iff p [= q "/\" r )
A1: q in {q,r} by TARSKI:def 2;
A2: r in {q,r} by TARSKI:def 2;
thus ( p is_less_than {q,r} implies p [= q "/\" r ) :: thesis: ( p [= q "/\" r implies p is_less_than {q,r} )
proof
assume A3: p is_less_than {q,r} ; :: thesis: p [= q "/\" r
then A4: p [= q by A1;
p [= r by A2, A3;
hence p [= q "/\" r by A4, FILTER_0:7; :: thesis: verum
end;
assume A5: p [= q "/\" r ; :: thesis: p is_less_than {q,r}
let a be Element of L; :: according to LATTICE3:def 16 :: thesis: ( a in {q,r} implies p [= a )
assume a in {q,r} ; :: thesis: p [= a
then A6: ( a = q or a = r ) by TARSKI:def 2;
A7: q "/\" r [= q by LATTICES:6;
r "/\" q [= r by LATTICES:6;
hence p [= a by A5, A6, A7, LATTICES:7; :: thesis: verum