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;
