let L be Lattice; :: thesis: for P being non empty ClosedSubset of L
for p, q being Element of
for p', q' being Element of st p = p' & q = q' holds
( p "\/" q = p' "\/" q' & p "/\" q = p' "/\" q' )
let P be non empty ClosedSubset of L; :: thesis: for p, q being Element of
for p', q' being Element of st p = p' & q = q' holds
( p "\/" q = p' "\/" q' & p "/\" q = p' "/\" q' )
let p, q be Element of ; :: thesis: for p', q' being Element of st p = p' & q = q' holds
( p "\/" q = p' "\/" q' & p "/\" q = p' "/\" q' )
let p', q' be Element of ; :: thesis: ( p = p' & q = q' implies ( p "\/" q = p' "\/" q' & p "/\" q = p' "/\" q' ) )
assume A1: ( p = p' & q = q' ) ; :: thesis: ( p "\/" q = p' "\/" q' & p "/\" q = p' "/\" q' )
consider o1, o2 being BinOp of P such that
A2: o1 = H₂(L) || P and
A3: o2 = H₃(L) || P and
A4: latt L,P = LattStr(# P,o1,o2 #) by Def16;
A5: [p',q'] in [:P,P:] by A4;
dom o1 = [:P,P:] by FUNCT_2:def 1;
hence p "\/" q = p' "\/" q' by A1, A2, A4, A5, FUNCT_1:70; :: thesis: p "/\" q = p' "/\" q'
dom o2 = [:P,P:] by FUNCT_2:def 1;
hence p "/\" q = p' "/\" q' by A1, A3, A4, A5, FUNCT_1:70; :: thesis: verum