let I be set ; :: thesis: for M being ManySortedSet of I
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 I; :: 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 :: thesis: for i being object st i in I holds
(meet |:SF:|) . i = (E (/\) T) . i

reconsider sf1 = SF as non empty SubsetFamily of M by A1;
let i be object ; :: thesis: ( i in I implies (meet |:SF:|) . i = (E (/\) T) . i )
assume A2: i in I ; :: thesis: (meet |:SF:|) . i = (E (/\) T) . i
ex Q being Subset-Family of (M . i) st
( Q = |:sf1:| . i & (meet |:sf1:|) . i = Intersect Q ) by A2, MSSUBFAM:def 1;
hence (meet |:SF:|) . i = meet (|:sf1:| . i) by A2, SETFAM_1:def 9
.= meet {(E . i),(T . i)} by A1, A2, Th20
.= (E . i) /\ (T . i) by SETFAM_1:11
.= (E (/\) T) . i by A2, PBOOLE:def 5 ;
:: thesis: verum

end;

hence meet |:SF:| = E (/\) T ; :: thesis: verum