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 Th33: p is U*-valued implies
(U-multiCat).(p^<*q*>) = (U-multiCat.p)^q
proof
set C=U-multiCat, g=U-concatenation, G=MultPlace g;
reconsider qq=q as FinSequence of U by Lm1;
per cases;
suppose p is empty; then reconsider e=p as empty set;
A1: (C.e)^q=q & C.(e^<*q*>) = C.(<*q*>);
C.(e^<*q*>)=G.(<*qq*>) by Th32 .= qq by Th31; hence thesis
by A1; end;
suppose A2: not p is empty; assume p is U*-valued; then reconsider
pp=p as non empty U*-valued FinSequence by A2;
reconsider ppp=pp as non empty FinSequence of U* by Lm1;
C.(pp^<*q*>)= G.(pp^<*qq*>) by Th32 .= g.(G.pp, qq) by Th31 .=
g.(C.ppp, q) by Th32 .= (C.p) ^ q by Th4; hence thesis;
end;
end;
