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;

theorem Th7: ::#Th7
A is D-prefix implies (D-multiCat).:(m-tuples_on A) is D-prefix
proof
reconsider f=D-concatenation as BinOp of (D*);
set F=D-multiCat, Y=F.:(m-tuples_on A); {} in D** by FINSEQ_1:49; then
A1: {} in dom F by FUNCT_2:def 1;
per cases;
suppose m=0; then
A2: Y=F.:({<*>A}) by FINSEQ_2:94 .= Im(F,{}) .= {F.{}}
by FUNCT_1:59, A1 .= {{}};
for x,y,d1,d2 being set st x in Y/\(D*) & y in Y/\(D*) &
d1 in D* & d2 in D* & f.(x,d1)=f.(y,d2) holds (x=y & d1=d2)
proof
let x,y,d1,d2 be  set; assume
A3: x in Y/\(D*) & y in Y/\(D*) & d1 in D* & d2 in D* & f.(x,d1)=f.(y,d2);
then
 x in Y & x in D* & y in Y & y in D* by XBOOLE_0:def 4;  then
A4: x={} & y={} by TARSKI:def 1,A2;
reconsider xx=x as Element of D* by A3;
reconsider yy=y as Element of D* by A3;
reconsider d11=d1 as Element of D* by A3;
reconsider d22=d2 as Element of D* by A3;
d11 = xx^d11 by A4,FINSEQ_1:34 .=
f.(yy,d22) by Lm10, A3 .=
{}^d22 by A4,Lm10 .= d22 by FINSEQ_1:34;
hence x=y & d1=d2 by A4;
end;
hence thesis by  Def10;
end;
suppose m<>0; then
consider k being Nat such that A5: m=k+1 by NAT_1:6;
set B=(k+1)-tuples_on A; B misses 0-tuples_on A by Lm5; then
B misses {{}} by Lm6; then A6: B\{{}}=B by XBOOLE_1:83;
assume A is D-prefix; then
A7:
(MultPlace(f)).:B is f-unambiguous by Lm19;
F.:B =  (F|(B\{{}})).:(B\{{}}) by RELAT_1:129, A6
.= ((MultPlace(f))|B) .: B by A6, Lm24
.= (MultPlace(f)).:B by RELAT_1:129;
hence thesis by A5, A7;
end;
end;
