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 U1 c= U2 & Y c= U1* & p is Y-valued & p<>{} &
Y is with_non-empty_elements implies U1-multiCat.p=U2-multiCat.p
proof
assume U1 c= U2; then reconsider U11=U1 as non empty Subset of U2;
assume Y c= U1*; then reconsider Y1=Y as Subset of U11*;
reconsider Y2 = Y1 as Subset of U2* by XBOOLE_1:1; assume
p is Y-valued; then reconsider p1=p as Y1-valued
FinSequence; reconsider p2=p1 as Y2-valued FinSequence;
assume p <> {} & Y is with_non-empty_elements; then
U1-multiCat.p1 <> {} & U2-multiCat.p2 <> {} by Lm52; hence thesis by Th52;
end;
