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 (U-multiCat.x is U1-valued & x in U**) implies
x is FinSequence of (U1*) ::#Th42
proof
set C=U-multiCat, f=U-concatenation, F=MultPlace f, D=U*;
{} null (U*) is U*-valued Relation; then
reconsider e={} as U*-valued FinSequence;
defpred P[Nat] means for p being ($1+1)-element U*-valued FinSequence
st C.p is U1-valued holds p is U1*-valued;
A1: P[0]
proof
let p be (0+1)-element U*-valued FinSequence;
reconsider ppp=p as (1+0)-element U*-valued FinSequence;
{ppp.1} \ U* ={}; then reconsider
p1=p.1 as Element of U* by ZFMISC_1:60;
A2: len p=1 by CARD_1:def 7; p={}^<*p.1*> by A2, FINSEQ_1:40; then
A3: C.p=(C.e)^(p1) by Th33 .= {}^p.1 .= p.1;
p is 1-element FinSequence of U* by Lm1; then
reconsider pp=p as 1-element Element of U**;
assume C.p is U1-valued; then reconsider u1=C.pp as FinSequence of U1
by Lm1; u1=p.1 by A3; then reconsider q=p.1 as Element of U1*;
<*q*> is FinSequence of U1*; hence thesis by A2, FINSEQ_1:40;
end;
A4: for n st P[n] holds P[n+1]
proof
let n; reconsider NN =
n+1 as non zero Element of NAT by ORDINAL1:def 12; assume
A5: P[n]; let p be (n+1+1)-element U*-valued FinSequence; assume
A6: C.p is U1-valued;
reconsider pp=p null p as (n+2)-element U*-valued FinSequence;
reconsider ppp=pp as (NN+1)-element U*-valued FinSequence;
reconsider pppp=ppp as (NN+1+0)-element U*-valued FinSequence;
reconsider p1=ppp|(Seg NN) as NN-element U*-valued FinSequence;
{pppp.(NN+1)} \ U* = {}; then
reconsider u=ppp.(NN+1) as Element of U* by ZFMISC_1:60;
A7: ppp \+\ (p1 ^ <*ppp.(NN+1)*>)={}; then
 p=p1^<*u*> by Th29; then
A8: C.p=(C.p1)^u by Th33; then rng (C.p) c= U1 &
rng (C.p1) c= rng (C.p) by A6, FINSEQ_1:29; then
reconsider q= C.p1 as
U1-valued FinSequence by XBOOLE_1:1, RELAT_1:def 19; q is U1-valued; then
reconsider p11=p1 as NN-element U1*-valued FinSequence by A5;
rng u c= rng (C.p) & rng (C.p) c= U1 by A8, FINSEQ_1:30, A6;
then u is U1-valued by  XBOOLE_1:1; then
u is FinSequence of U1 by Lm1; then reconsider uu=u as Element of U1*;
p11^<*uu*> is U1*-valued; hence thesis by A7, Th29;
end;
A9: for n holds P[n] from NAT_1:sch 2(A1,A4); assume
A10: C.x is U1-valued & x in U**;
per cases;
suppose x is empty; then reconsider xx=x as empty set;
xx null (U1*) is (U1*)-valued FinSequence;
hence thesis by Lm1;
end;
suppose not x is empty; then reconsider xx=x as non empty U*-valued
FinSequence by A10; consider m such that
A11: len xx=m+1 by NAT_1:6; xx null {} is (m+1)-element by A11; then reconsider
xxx=xx as (m+1)-element U*-valued FinSequence;
xxx is U1*-valued by A10, A9; hence thesis by Lm1;
end;
end;
