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;
