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 Th63: B is c=directed & A is_finer_than B implies
A\/B is c=directed
proof
assume
A1: B is c=directed & A is_finer_than B;
reconsider BB=B as
c=directed set by A1; reconsider X=A\/BB as non empty set;
now
let a, b be set; assume a in X; then consider aa being set such that
A2: aa in B & a c= aa by SETFAM_1:def 2, A1, Th62; assume
b in X; then consider bb being set such that
A3: bb in B & b c= bb by SETFAM_1:def 2, A1, Th62;
consider cc being set such that
A4: aa\/bb c= cc & cc in B by A1, A2, A3, COHSP_1:5; take cc;
a\/b c= aa\/bb by A2, A3, XBOOLE_1:13; hence a\/b c= cc by A4;
thus cc in X by A4, XBOOLE_0:def 3;
end; hence A\/B is c=directed by COHSP_1:6;
end;
