reserve k,m,n for Nat, kk,mm,nn for Element of NAT, X,Y,x,y,z for set;
reserve S,S1,S2 for Language, s,s1,s2 for Element of S;
reserve l,l1,l2 for literal Element of S, a for ofAtomicFormula Element of S,
r for relational Element of S, w,w1,w2 for string of S,
t,t1,t2 for termal string of S, tt,tt1, tt2 for Element of AllTermsOf S;
reserve phi0 for 0wff string of S;

theorem X/\LettersOf S2 is infinite implies ex S1 st
(OwnSymbolsOf S1 = X/\OwnSymbolsOf S2 & S2 is S1-extending)
proof
set L2=LettersOf S2, O2=OwnSymbolsOf S2, a2=the adicity of S2, E2=TheEqSymbOf
S2, N2=TheNorSymbOf S2, SS2=AllSymbolsOf S2;
reconsider X1=SS2/\X as Subset of SS2; reconsider
N22=N2, E22=E2 as Element of SS2; {E22,N22} is Subset of SS2; then
reconsider X2={E2,N2} as non empty Subset of SS2; set SS1=X1\/X2;
assume X/\L2 is infinite; then reconsider L11=X/\L2 as infinite set;
L11 c= X/\SS2 null {E2,N2} by XBOOLE_1:26;
then reconsider SS11=SS1 as infinite Subset of SS2;
reconsider AS11=SS11\{N2} as infinite Subset of SS11;
E2 in X2 null X1 & not E2 in {N2} by TARSKI:def 1, def 2; then
reconsider E1=E2 as Element of AS11 by XBOOLE_0:def 5;
N2 in X2 null X1 by TARSKI:def 2;
then reconsider N1=N2 as Element of SS11;
reconsider D1=SS11\{N1} as infinite Subset of SS2\{N2} by XBOOLE_1:33;
rng (a2|D1) c= INT & dom (a2|D1)=D1 by PARTFUN1:def 2; then
reconsider a1=a2|D1 as Function of SS11\{N1}, INT by FUNCT_2:2;
reconsider a11=a2|D1 as Subset of a2;
set S1=Language-like (# SS11, E1 qua Element of SS11, N1, a1 #),
O1=OwnSymbolsOf S1, L1=LettersOf S1;
reconsider IT=S1 as non degenerated Language-like by Def44;
A1: L1 = a2"{0} /\ D1 by FUNCT_1:70 .=
L2 /\ SS11 \ {N1} by XBOOLE_1:49 .=
L2 /\ (SS2/\X) \/ L2/\{E2,N2} \ {N2} by XBOOLE_1:23 .=
L2 null SS2 /\ X\/L2/\{E2,N2}\{N2} by XBOOLE_1:16
.= L11\/(L2/\{E2,N2})\{N2};
a1.E1 \+\ a2.E1 = {}; then a1.E1=a2.E2 &
a2.E2=-2 by Def23, FOMODEL0:29; then
(the adicity of IT).(TheEqSymbOf IT)=-2 &
LettersOf IT is infinite by A1; then reconsider IT as Language
by Def23; take IT; SS1\X2=X1\X2 \/ (X2\X2) by XBOOLE_1:42 .=
O2/\X by XBOOLE_1:49;
hence OwnSymbolsOf IT = X/\O2; thus thesis;
end;
