theorem Th61:
for X being set holds
A is_finer_than B & X is_finer_than Y implies A\/X is_finer_than B\/Y
proof
let X be set;
set LH=A\/X, RH=B\/Y; assume
A1: A is_finer_than B & X is_finer_than Y;
now
let Z be set; assume
A2: Z in LH;
per cases by XBOOLE_0:def 3, A2;
suppose
Z in A; then consider b being set such that
A3: b in B & Z c= b by SETFAM_1:def 2, A1; take b;
thus b in RH by A3, XBOOLE_0:def 3; thus Z c= b by A3;
end;
suppose Z in X; then consider y being set such that
A4: y in Y & Z c= y by SETFAM_1:def 2, A1; take y;
thus y in RH by A4, XBOOLE_0:def 3; thus Z c= y by A4;
end;
end; hence thesis by SETFAM_1:def 2;
end;
