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 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;
