theorem Y is c=directed implies for X being finite Subset of union Y
ex y st y in Y & X c= y ::#COHSP_1:13 with shorter proof
proof
set F=Y; assume
A1: F is c=directed; let X be finite Subset of union F;
X/\union F = union INTERSECTION ({X},F) by SETFAM_1:25; then
reconsider FF=INTERSECTION({X}, F) as finite Subset-Family of X by Th60;
A2: X null union F = union FF by SETFAM_1:25;
F\/FF is c=directed & FF null F c= F\/FF by Th63, Th64, A1; then
consider a being set such that
A3: union FF c= a & a in F\/FF by COHSP_1:def 4;
a in F or (a in FF & FF is_finer_than F) by A3, Th64, XBOOLE_0:def 3;
then consider b being set such that
A4: b in F & a c= b by SETFAM_1:def 2; take b;
thus b in F & X c= b by A2, A3, A4;
end;
