let I be I_Lattice; :: thesis: for FI being Filter of I
for i, j, k being Element of I st i => k in FI & j => k in FI holds
(i "\/" j) => k in FI
let FI be Filter of I; :: thesis: for i, j, k being Element of I st i => k in FI & j => k in FI holds
(i "\/" j) => k in FI
let i, j, k be Element of I; :: thesis: ( i => k in FI & j => k in FI implies (i "\/" j) => k in FI )
assume that
A1: i => k in FI and
A2: j => k in FI ; :: thesis: (i "\/" j) => k in FI
A3: (i => k) "/\" (j => k) [= (i "\/" j) => k by Th56;
(i => k) "/\" (j => k) in FI by A1, A2, FILTER_0:8;
hence (i "\/" j) => k in FI by A3, FILTER_0:9; :: thesis: verum