let I be set ; :: thesis: for M being ManySortedSet of
for SF being SubsetFamily of M
for E, T being Element of Bool M st SF = {E,T} holds
meet |:SF:| = E /\ T
let M be ManySortedSet of ; :: thesis: for SF being SubsetFamily of M
for E, T being Element of Bool M st SF = {E,T} holds
meet |:SF:| = E /\ T
let SF be SubsetFamily of M; :: thesis: for E, T being Element of Bool M st SF = {E,T} holds
meet |:SF:| = E /\ T
let E, T be Element of Bool M; :: thesis: ( SF = {E,T} implies meet |:SF:| = E /\ T )
assume A1: SF = {E,T} ; :: thesis: meet |:SF:| = E /\ T

now

let i be set ; :: thesis: ( i in I implies (meet |:SF:|) . i = (E /\ T) . i )
assume A2: i in I ; :: thesis: (meet |:SF:|) . i = (E /\ T) . i
reconsider sf1 = SF as non empty SubsetFamily of M by A1;
consider Q being Subset-Family of (M . i) such that
A3: Q = |:sf1:| . i and
A4: (meet |:sf1:|) . i = Intersect Q by A2, MSSUBFAM:def 2;
Q <> {} by A2, A3;
hence (meet |:SF:|) . i = meet (|:sf1:| . i) by A3, A4, SETFAM_1:def 10
.= meet {(E . i),(T . i)} by A1, A2, Th21
.= (E . i) /\ (T . i) by SETFAM_1:12
.= (E /\ T) . i by A2, PBOOLE:def 8 ;
:: thesis: verum

end;

hence meet |:SF:| = E /\ T by PBOOLE:3; :: thesis: verum