let L be Lattice; :: thesis: for p, q being Element of L
for F being Filter of L
for p9, q9 being Element of (latt F) st p = p9 & q = q9 holds
( p "\/" q = p9 "\/" q9 & p "/\" q = p9 "/\" q9 )
let p, q be Element of L; :: thesis: for F being Filter of L
for p9, q9 being Element of (latt F) st p = p9 & q = q9 holds
( p "\/" q = p9 "\/" q9 & p "/\" q = p9 "/\" q9 )
let F be Filter of L; :: thesis: for p9, q9 being Element of (latt F) st p = p9 & q = q9 holds
( p "\/" q = p9 "\/" q9 & p "/\" q = p9 "/\" q9 )
let p9, q9 be Element of (latt F); :: thesis: ( p = p9 & q = q9 implies ( p "\/" q = p9 "\/" q9 & p "/\" q = p9 "/\" q9 ) )
assume A1: ( p = p9 & q = q9 ) ; :: thesis: ( p "\/" q = p9 "\/" q9 & p "/\" q = p9 "/\" q9 )
consider o1, o2 being BinOp of F such that
A2: o1 = the L_join of L || F and
A3: o2 = the L_meet of L || F and
A4: latt F = LattStr(# F,o1,o2 #) by Def9;
dom o1 = [:F,F:] by FUNCT_2:def 1;
then [p,q] in dom o1 by A1, A4, ZFMISC_1:87;
hence p "\/" q = p9 "\/" q9 by A1, A2, A4, FUNCT_1:47; :: thesis: p "/\" q = p9 "/\" q9
dom o2 = [:F,F:] by FUNCT_2:def 1;
then [p,q] in dom o2 by A1, A4, ZFMISC_1:87;
hence p "/\" q = p9 "/\" q9 by A1, A3, A4, FUNCT_1:47; :: thesis: verum