reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;
reserve f for BinOp of D;
reserve a,a1,a2,b,b1,b2,A,B,C,X,Y,Z,x,x1,x2,y,y1,y2,z for set,
U,U1,U2,U3 for non empty set, u,u1,u2 for Element of U,
P,Q,R for Relation, f,f1,f2,g,g1,g2 for Function,
k,m,n for Nat, kk,mm,nn for Element of NAT, m1, n1 for non zero Nat,
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;

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;
