reserve k,m,n for Nat, kk,mm,nn for Element of NAT, A,B,X,Y,Z,x,y,z for set,
S, S1, S2 for Language, s for (Element of S), w,w1,w2 for (string of S),
U,U1,U2 for non empty set, f,g for Function, p,p1,p2 for FinSequence;
reserve u,u1,u2 for Element of U, t for termal string of S,
I for (S,U)-interpreter-like Function,
l, l1, l2 for literal (Element of S), m1, n1 for non zero Nat,
phi0 for 0wff string of S, psi,phi,phi1,phi2 for wff string of S;

theorem Th10: AtomicFormulasOf S is S-prefix
proof
set AF=AtomicFormulasOf S, SS=AllSymbolsOf S, TT=AllTermsOf S, C=S-multiCat;
now
let p1,q1,p2,q2 be SS-valued FinSequence;
assume p1 in AF; then consider phi1 being string of S such that
A1: p1=phi1 & phi1 is 0wff;
assume p2 in AF; then consider phi2 being string of S such that
A2: p2=phi2 & phi2 is 0wff; consider r1 being relational Element of S,
T1 being |.ar r1.|-element Element of TT* such that
A3: phi1=<*r1*>^(C.T1) by A1;
consider r2 being relational Element of S,
T2 being |.ar r2.|-element Element of TT* such that
A4: phi2=<*r2*>^(C.T2) by A2; assume p1^q1=p2^q2; then
A5: <*r1*>^((C.T1)^q1)=(<*r2*>^(C.T2))^q2 by A1, A2, A3, A4, FINSEQ_1:32 .=
<*r2*>^((C.T2)^q2) by FINSEQ_1:32; then
A6: r1 = (<*r2*>^((C.T2)^q2)).1
by FINSEQ_1:41 .= r2 by FINSEQ_1:41;
set n=|.ar r1.|, nT=n-tuples_on TT;
reconsider T11=T1, T22=T2 as Element of nT by FOMODEL0:16, A6;
reconsider P=C.:(nT) as SS-prefix set;
A7: (SS*\{{}})* c= SS** & TT*c=(SS*\{{}})*; then
 T1 in SS** & T2 in SS** & dom C=SS**
by  FUNCT_2:def 1; then
A8: C.T11 in P & C.T22 in P by FUNCT_1:def 6;
reconsider T111=T1, T222=T2 as Element of SS**
by A7;
(C.T111)^q1=(C.T222)^q2 by A6, FINSEQ_1:33, A5;
hence p1=p2 & q1=q2 by A1, A2, A8, FOMODEL0:def 19, A3, A4, A6;
end;
then AF is SS-prefix; hence thesis;
end;
