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 A is D-prefix implies D-multiCat is (m-tuples_on A)-one-to-one
proof
set f=D-concatenation, F=D-multiCat, Z=m-tuples_on A; assume
A1: A is D-prefix;
per cases;
suppose m=0;
then Z=Funcs(Seg 0,A) by Lm7 .= {{}} by FUNCT_5:57;
then F|Z is one-to-one;
hence thesis;
end;
suppose m<>0; then consider k being Nat such that A2: m=k+1 by NAT_1:6;
reconsider kk=k+1 as Element of NAT by ORDINAL1:def 12;
set ZZ=kk-tuples_on A;
(MultPlace(f)) is ZZ-one-to-one by A1, Lm21; then
A3: (MultPlace(f)) | ZZ is one-to-one;
len {} = 0; then not {} in ZZ by FINSEQ_2:132; then {{}}
misses ZZ by ZFMISC_1:50; then ZZ\{{}} = ZZ by XBOOLE_1:83;
then F|Z is one-to-one by Lm24, A3,A2; hence thesis;
end;
end;
