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;
