let I be I_Lattice; :: thesis: for FI being Filter of I
for i, j, k being Element of I st i => j in FI & i => 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 => j in FI & i => k in FI holds
i => (j "/\" k) in FI
let i, j, k be Element of I; :: thesis: ( i => j in FI & i => k in FI implies i => (j "/\" k) in FI )
assume that
A1: i => j in FI and
A2: i => k in FI ; :: thesis: i => (j "/\" k) in FI
A3: (i => j) "/\" (i => k) [= i => (j "/\" k) by Th57;
(i => j) "/\" (i => k) in FI by A1, A2, FILTER_0:8;
hence i => (j "/\" k) in FI by A3, FILTER_0:9; :: thesis: verum