let S be non empty RelStr ; :: thesis: for T being complete LATTICE
for F being non empty Subset of (T |^ the carrier of S)
for i being Element of S holds (sup F) . i = "\/" { (f . i) where f is Element of (T |^ the carrier of S) : f in F } ,T
let T be complete LATTICE; :: thesis: for F being non empty Subset of (T |^ the carrier of S)
for i being Element of S holds (sup F) . i = "\/" { (f . i) where f is Element of (T |^ the carrier of S) : f in F } ,T
let F be non empty Subset of (T |^ the carrier of S); :: thesis: for i being Element of S holds (sup F) . i = "\/" { (f . i) where f is Element of (T |^ the carrier of S) : f in F } ,T
let i be Element of S; :: thesis: (sup F) . i = "\/" { (f . i) where f is Element of (T |^ the carrier of S) : f in F } ,T
reconsider SYT = the carrier of S --> T as RelStr-yielding non-Empty ManySortedSet of ;
reconsider SYT = SYT as RelStr-yielding non-Empty reflexive-yielding ManySortedSet of ;
A1: T |^ the carrier of S = product SYT by YELLOW_1:def 5;
reconsider X = F as Subset of (product SYT) by YELLOW_1:def 5;
A2: pi X,i = { (f . i) where f is Element of (T |^ the carrier of S) : f in F }

proof

thus pi X,i c= { (f . i) where f is Element of (T |^ the carrier of S) : f in F } :: according to XBOOLE_0:def 10 :: thesis: { (f . i) where f is Element of (T |^ the carrier of S) : f in F } c= pi X,i

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in pi X,i or a in { (f . i) where f is Element of (T |^ the carrier of S) : f in F } )
assume a in pi X,i ; :: thesis: a in { (f . i) where f is Element of (T |^ the carrier of S) : f in F }
then consider g being Function such that
A3: ( g in X & a = g . i ) by CARD_3:def 6;
thus a in { (f . i) where f is Element of (T |^ the carrier of S) : f in F } by A3; :: thesis: verum

end;

thus { (f . i) where f is Element of (T |^ the carrier of S) : f in F } c= pi X,i :: thesis: verum

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in { (f . i) where f is Element of (T |^ the carrier of S) : f in F } or a in pi X,i )
assume a in { (f . i) where f is Element of (T |^ the carrier of S) : f in F } ; :: thesis: a in pi X,i
then consider g being Element of (T |^ the carrier of S) such that
A4: ( a = g . i & g in F ) ;
thus a in pi X,i by A4, CARD_3:def 6; :: thesis: verum

end;

end;

for i being Element of S holds SYT . i is complete LATTICE by FUNCOP_1:13;
then (sup F) . i = "\/" (pi X,i),(SYT . i) by A1, WAYBEL_3:32
.= "\/" { (f . i) where f is Element of (T |^ the carrier of S) : f in F } ,T by A2, FUNCOP_1:13 ;
hence (sup F) . i = "\/" { (f . i) where f is Element of (T |^ the carrier of S) : f in F } ,T ; :: thesis: verum