let L be Lattice; :: thesis: for u being Element of L
for A being non empty set
for B being Finite_Subset of A
for f being Function of A,the carrier of L st B <> {} & ( for x being Element of A st x in B holds
u [= f . x ) holds
u [= FinMeet B,f
let u be Element of L; :: thesis: for A being non empty set
for B being Finite_Subset of A
for f being Function of A,the carrier of L st B <> {} & ( for x being Element of A st x in B holds
u [= f . x ) holds
u [= FinMeet B,f
let A be non empty set ; :: thesis: for B being Finite_Subset of A
for f being Function of A,the carrier of L st B <> {} & ( for x being Element of A st x in B holds
u [= f . x ) holds
u [= FinMeet B,f
let B be Finite_Subset of A; :: thesis: for f being Function of A,the carrier of L st B <> {} & ( for x being Element of A st x in B holds
u [= f . x ) holds
u [= FinMeet B,f
let f be Function of A,the carrier of L; :: thesis: ( B <> {} & ( for x being Element of A st x in B holds
u [= f . x ) implies u [= FinMeet B,f )
assume that
A1: B <> {} and
A2: for x being Element of A st x in B holds
u [= f . x ; :: thesis: u [= FinMeet B,f
reconsider f' = f as Function of A,the carrier of (L .: ) ;
reconsider u' = u as Element of (L .: ) ;
for x being Element of A st x in B holds
f' . x [= u' by A2, Th53;
then FinJoin B,f' [= u' by A1, Th47;
hence u [= FinMeet B,f by Th54; :: thesis: verum