theorem Th18: 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 addw (D,num) c= Z & Z is D-consistent & rng num = ExFormulasOf S
holds Z is S-witnessed
proof
let X be functional set;set L=LettersOf S,F=S-firstChar,EF=ExFormulasOf S;
let num be sequence of  EF; set f=(D,num) addw X, Y=X addw (D,num),
SS=AllSymbolsOf S; X\Y ={}; then
A1: X c= Y by XBOOLE_1:37; assume
A2: 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; assume
A3: Y c= Z & Z is D-consistent; then X c= Z & Z is D-consistent
by A1, XBOOLE_1:1; then
A4: X is D-consistent; assume
A5: rng num = EF; set strings=SS*\{{}};
for l1, phi1 st <*l1*>^phi1 in Z ex l2 st (
(l1,l2)-SymbolSubstIn phi1 in Z & not l2 in rng phi1)
proof
let l1, phi1; set phi=<*l1*>^phi1;
phi=<*l1*>^phi1^{} & not phi is 0wff; then
A6: l1=F.phi & phi1=head phi by FOMODEL2:23;
phi in EF; then
reconsider phii=phi as Element of EF; consider x being object such that
A7: x in dom num & num.x=phii by A5, FUNCT_1:def 3;
reconsider mm=x as Element of NAT by A7;
::#this works because of redefine func dom in relset_1
reconsider MM=mm+1 as Element of NAT by ORDINAL1:def 12;
reconsider Xm=f.mm as functional set;
set no=SymbolsOf (strings/\(f.mm\/{phi1})); reconsider T=strings/\{phi1}
as FinSequence-membered finite Subset of {phi1};
reconsider t=SymbolsOf T as finite set;
reconsider i=L\SymbolsOf (f.mm/\strings) as infinite Subset of L
by Th17, A2, A4;
A8: no= SymbolsOf ((strings/\f.mm)\/(strings/\{phi1})) by XBOOLE_1:23 .=
SymbolsOf (strings/\f.mm) \/ SymbolsOf T by FOMODEL0:47; then
L\no=i\t by XBOOLE_1:41;
then reconsider yes=L\no as non empty Subset of L;
set ll2=the Element of yes;
reconsider l2=ll2 as literal Element of S by TARSKI:def 3;
set psi1=(l1,l2)-SymbolSubstIn phi1;
dom f=NAT by FUNCT_2:def 1; then
A9: f.mm in rng f & f.MM in rng f by FUNCT_1:def 3; then
f.mm c= Y by ZFMISC_1:74; then
A10: f.mm c= Z by A3, XBOOLE_1:1;
assume phi in Z; then {phi} c= Z by ZFMISC_1:31; then
f.mm \/ {phi} c= Z by A10, XBOOLE_1:8;
then f.mm \/ {phi} is D-consistent by A3; then
f.mm \/ {(l1,l2)-SymbolSubstIn phi1}  =
(D,phii) AddAsWitnessTo f.mm by Def66, A6 .=
f.(mm+1) by Def71, A7; then {psi1} null (f.mm) c= f.MM; then
psi1 in f.MM by ZFMISC_1:31; then
A11: psi1 in Y by TARSKI:def 4, A9; take l2;
thus (l1,l2)-SymbolSubstIn phi1 in Z by A3, A11;
not l2 in no by XBOOLE_0:def 5; then
not l2 in SymbolsOf {phi1} by A8, XBOOLE_0:def 3;
hence thesis by FOMODEL0:45;
end; hence Z is S-witnessed;
end;
