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

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