reserve k,m,n for Nat, kk,mm,nn for Element of NAT,
 U,U1,U2 for non empty set,
 A,B,X,Y,Z, x,x1,x2,y,z for set,
 S for Language, s, s1, s2 for Element of S,
f,g for Function, w for string of S, tt,tt1,tt2 for Element of AllTermsOf S,
psi,psi1,psi2,phi,phi1,phi2 for wff string of S, u,u1,u2 for Element of U,
Phi,Phi1,Phi2 for Subset of AllFormulasOf S, t,t1,t2,t3 for termal string of
S,
r for relational Element of S, a for ofAtomicFormula Element of S,
l, l1, l2 for literal Element of S, p for FinSequence,
m1, n1 for non zero Nat, S1, S2 for Language;
reserve D,D1,D2,D3 for RuleSet of S, R for Rule of S,
Seqts,Seqts1,Seqts2 for Subset of S-sequents,
seqt,seqt1,seqt2 for Element of S-sequents,
SQ,SQ1,SQ2 for S-sequents-like set, Sq,Sq1,Sq2 for S-sequent-like object;
reserve H,H1,H2,H3 for S-premises-like set;
reserve M,K,K1,K2 for isotone RuleSet of S;
 reserve D,E,F for (RuleSet of S), D1 for 1-ranked 0-ranked RuleSet of S;

theorem Th17: for X being functional set,
num being sequence of  ExFormulasOf S st
D is isotone & R#1(S) in D & R#8(S) in D & R#2(S) in D & R#5(S) in D &
LettersOf S\SymbolsOf (X/\((AllSymbolsOf S)*\{{}})) is infinite &
X is D-consistent
holds
((D,num) addw X).k c= ((D,num) addw X).(k+m) &
LettersOf S\SymbolsOf (((D,num) addw X).m/\((AllSymbolsOf S)*\{{}}))
is infinite & ((D,num) addw X).m is D-consistent
proof
let X be functional set; set L=LettersOf S,F=S-firstChar,FF=
AllFormulasOf S,SS=AllSymbolsOf S,strings=SS*\{{}},EF=ExFormulasOf S;
let num be sequence of  EF; set f=(D,num) addw X; assume
A1: D is isotone & R#1(S) in D & R#8(S) in D & R#2(S) in D & R#5(S) in D;
assume
A2: L\SymbolsOf (X/\strings) is infinite & X is D-consistent;
defpred P[Nat] means f.k c= f.(k+$1) &
L\SymbolsOf (f.$1 /\ strings) is infinite & f.$1 is D-consistent;
A3: P[0] by A2, Def71;
A4: for m st P[m] holds P[m+1]
proof
let m; reconsider mk=k+m, MM=m+1, mm=m as Element of NAT by ORDINAL1:def 12;
reconsider phii=num.mm as Element of EF;
reconsider phi=num.mm as exal wff string of S by TARSKI:def 3;
reconsider phi1=head phi as wff string of S;
reconsider l1=F.phi as literal Element of S;
A5: phi=<*l1*>^phi1^(tail phi) by FOMODEL2:23 .= <*l1*>^phi1;
reconsider fmk=(D, num.mk) AddAsWitnessTo (f.mk) as Subset of (f.mk\/FF);
reconsider fmm=(D, num.mm) AddAsWitnessTo (f.mm) as Subset of (f.mm\/FF);
 f.mk \ fmk = {}; then
 f.mk c= fmk by XBOOLE_1:37; then
A6: f.mk c= f.(mk+1) & f.MM=fmm by Def71; assume
A7: P[m];
hence f.k c= f.(k+(m+1)) by A6, XBOOLE_1:1; (f.mm)\fmm={}; then
reconsider fm=f.mm as functional Subset of fmm by XBOOLE_1:37;
reconsider sm=fm/\strings as Subset of (fmm/\strings) by XBOOLE_1:26;
reconsider t=fmm\(f.mm) as trivial set;
reconsider i=L\SymbolsOf sm as infinite set by A7;
reconsider T=t/\strings as functional finite FinSequence-membered set;
fmm=fm \/ t by XBOOLE_1:45; then
SymbolsOf (fmm/\strings)=
SymbolsOf (sm\/T) by XBOOLE_1:23 .=
SymbolsOf sm \/ SymbolsOf T by FOMODEL0:47; then
L\SymbolsOf (fmm/\strings)=i\SymbolsOf T by XBOOLE_1:41;
hence L\SymbolsOf (f.(m+1)/\strings) is infinite by Def71;
reconsider LF=L\SymbolsOf(strings/\(fm\/{head phii})) as Subset of L;
per cases;
suppose
A8: fm \/ {phii} is D-consistent & LF<>{}; then
reconsider LF as non empty Subset of L; set ll2=the Element of LF;
reconsider l2=ll2 as literal Element of S
by TARSKI:def 3; not ll2 in SymbolsOf(strings/\(fm\/{head phii}))
by XBOOLE_0:def 5; then fm\/{<*l1*>^phi1} is D-consistent
& l2 is (fm\/{phi1})-absent by A8, A5; then
A9: fm\/{(l1,l2)-SymbolSubstIn phi1} is D-consistent by Lm49, A1;
thus thesis by A8, Def66, A9, A6;
end;
suppose not (fm \/ {phii} is D-consistent & LF<>{});
hence thesis by A7, A6, Def66;
end;
end;
for n holds P[n] from NAT_1:sch 2(A3, A4); hence thesis;
end;
