let I be non empty set ; :: thesis: for J being non-Empty TopStruct-yielding ManySortedSet of I
for i1, i2 being Element of I
for xi1 being Element of (J . i1)
for Ai2 being Subset of (J . i2) st Ai2 <> [#] (J . i2) holds
( (proj (J,i1)) " {xi1} c= (proj (J,i2)) " Ai2 iff ( i1 = i2 & xi1 in Ai2 ) )
let J be non-Empty TopStruct-yielding ManySortedSet of I; :: thesis: for i1, i2 being Element of I
for xi1 being Element of (J . i1)
for Ai2 being Subset of (J . i2) st Ai2 <> [#] (J . i2) holds
( (proj (J,i1)) " {xi1} c= (proj (J,i2)) " Ai2 iff ( i1 = i2 & xi1 in Ai2 ) )
let i1, i2 be Element of I; :: thesis: for xi1 being Element of (J . i1)
for Ai2 being Subset of (J . i2) st Ai2 <> [#] (J . i2) holds
( (proj (J,i1)) " {xi1} c= (proj (J,i2)) " Ai2 iff ( i1 = i2 & xi1 in Ai2 ) )
let xi1 be Element of (J . i1); :: thesis: for Ai2 being Subset of (J . i2) st Ai2 <> [#] (J . i2) holds
( (proj (J,i1)) " {xi1} c= (proj (J,i2)) " Ai2 iff ( i1 = i2 & xi1 in Ai2 ) )
let Ai2 be Subset of (J . i2); :: thesis: ( Ai2 <> [#] (J . i2) implies ( (proj (J,i1)) " {xi1} c= (proj (J,i2)) " Ai2 iff ( i1 = i2 & xi1 in Ai2 ) ) )
reconsider Ai29 = Ai2 as Subset of ((Carrier J) . i2) by YELLOW_6:2;
i2 in I ;
then A1: i2 in dom (Carrier J) by PARTFUN1:def 2;
assume Ai2 <> [#] (J . i2) ; :: thesis: ( (proj (J,i1)) " {xi1} c= (proj (J,i2)) " Ai2 iff ( i1 = i2 & xi1 in Ai2 ) )
then A2: Ai29 <> (Carrier J) . i2 by YELLOW_6:2;
xi1 in the carrier of (J . i1) ;
then A3: xi1 in (Carrier J) . i1 by YELLOW_6:2;
i1 in I ;
then ( product (Carrier J) <> {} & i1 in dom (Carrier J) ) by PARTFUN1:def 2;
then ( (proj ((Carrier J),i1)) " {xi1} c= (proj ((Carrier J),i2)) " Ai29 iff ( i1 = i2 & xi1 in Ai29 ) ) by A1, A3, A2, Th7;
then ( (proj (J,i1)) " {xi1} c= (proj ((Carrier J),i2)) " Ai2 iff ( i1 = i2 & xi1 in Ai29 ) ) by WAYBEL18:def 4;
hence ( (proj (J,i1)) " {xi1} c= (proj (J,i2)) " Ai2 iff ( i1 = i2 & xi1 in Ai2 ) ) by WAYBEL18:def 4; :: thesis: verum