let L be non empty transitive RelStr ; :: thesis: for S being non empty join-closed Subset of L
for x, y being Element of S st ex_sup_of {x,y},L holds
( ex_sup_of {x,y}, subrelstr S & "\/" ({x,y},(subrelstr S)) = "\/" ({x,y},L) )
let S be non empty join-closed Subset of L; :: thesis: for x, y being Element of S st ex_sup_of {x,y},L holds
( ex_sup_of {x,y}, subrelstr S & "\/" ({x,y},(subrelstr S)) = "\/" ({x,y},L) )
let x, y be Element of S; :: thesis: ( ex_sup_of {x,y},L implies ( ex_sup_of {x,y}, subrelstr S & "\/" ({x,y},(subrelstr S)) = "\/" ({x,y},L) ) )
A1: x is Element of (subrelstr S) by YELLOW_0:def 15;
A2: y is Element of (subrelstr S) by YELLOW_0:def 15;
assume A3: ex_sup_of {x,y},L ; :: thesis: ( ex_sup_of {x,y}, subrelstr S & "\/" ({x,y},(subrelstr S)) = "\/" ({x,y},L) )
subrelstr S is non empty full join-inheriting SubRelStr of L by Def2;
then "\/" ({x,y},L) in the carrier of (subrelstr S) by A1, A2, A3, YELLOW_0:def 17;
hence ( ex_sup_of {x,y}, subrelstr S & "\/" ({x,y},(subrelstr S)) = "\/" ({x,y},L) ) by A1, A2, A3, YELLOW_0:66; :: thesis: verum