let L be sup-Semilattice; :: thesis: for C being non empty Subset of L st ( for x, y being Element of L st x in C & y in C & not x <= y holds
y <= x ) holds
for Y being non empty finite Subset of C holds "\/" (Y,L) in Y
let C be non empty Subset of L; :: thesis: ( ( for x, y being Element of L st x in C & y in C & not x <= y holds
y <= x ) implies for Y being non empty finite Subset of C holds "\/" (Y,L) in Y )
assume A1: for x, y being Element of L st x in C & y in C & not x <= y holds
y <= x ; :: thesis: for Y being non empty finite Subset of C holds "\/" (Y,L) in Y
defpred S₁[ set ] means ( "\/" ($1,L) in $1 & ex_sup_of $1,L );
A2: for B1, B2 being non empty Element of Fin C st S₁[B1] & S₁[B2] holds
S₁[B1 \/ B2]

proof

let B1, B2 be non empty Element of Fin C; :: thesis: ( S₁[B1] & S₁[B2] implies S₁[B1 \/ B2] )
assume A3: ( S₁[B1] & S₁[B2] ) ; :: thesis: S₁[B1 \/ B2]
( B1 c= C & B2 c= C ) by FINSUB_1:def 5;
then ( "\/" (B1,L) <= "\/" (B2,L) or "\/" (B2,L) <= "\/" (B1,L) ) by A1, A3;
then A4: ( ("\/" (B1,L)) "\/" ("\/" (B2,L)) = "\/" (B1,L) or ("\/" (B1,L)) "\/" ("\/" (B2,L)) = "\/" (B2,L) ) by YELLOW_0:24;
"\/" ((B1 \/ B2),L) = ("\/" (B1,L)) "\/" ("\/" (B2,L)) by A3, YELLOW_2:3;
hence S₁[B1 \/ B2] by A3, A4, XBOOLE_0:def 3, YELLOW_2:3; :: thesis: verum

end;

let Y be non empty finite Subset of C; :: thesis: "\/" (Y,L) in Y
A5: Y in Fin C by FINSUB_1:def 5;
A6: for x being Element of C holds S₁[{x}]

proof

let x be Element of C; :: thesis: S₁[{x}]
"\/" ({x},L) = x by YELLOW_0:39;
hence S₁[{x}] by TARSKI:def 1, YELLOW_0:38; :: thesis: verum

end;

for B being non empty Element of Fin C holds S₁[B] from SETWISEO:sch 3(A6, A2);
hence "\/" (Y,L) in Y by A5; :: thesis: verum