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;
reserve D2 for 2-ranked RuleSet of S;
reserve C for countable Language, phi for wff string of C;

theorem for X2 being countable Subset of AllFormulasOf S2, I2 being Element of
U-InterpretersOf S2 st X2 is I2-satisfied ex
U1 being countable non empty set, I1 being Element of U1-InterpretersOf S2
st X2 is I1-satisfied
proof
set FF2=AllFormulasOf S2, L2=LettersOf S2; let X2 be
countable Subset of FF2; let I2 be Element of U-InterpretersOf S2; assume
A1: X2 is I2-satisfied;  set L = the denumerable Subset of L2;
reconsider SS1=L\/SymbolsOf X2 as denumerable set;
L2/\SS1 = L null L2 \/ (L2/\SymbolsOf X2) by XBOOLE_1:23; then
consider S1 such that
A2: OwnSymbolsOf S1 = SS1/\OwnSymbolsOf S2 & S2 is S1-extending
by FOMODEL1:18; AllSymbolsOf S1 =
OwnSymbolsOf S1 \/ (AllSymbolsOf S1/\{TheEqSymbOf S1, TheNorSymbOf S1})
& OwnSymbolsOf S1 is countable by A2; then
reconsider S11=S1 as countable Language by ORDERS_4:def 2;
reconsider S22=S2 as S11-extending Language by A2;
set II11=U-InterpretersOf S11, II22=U-InterpretersOf S22,
O11=OwnSymbolsOf S11, FF11=AllFormulasOf S11, O22=OwnSymbolsOf S22,
a11=the adicity of S11, a22=the adicity of S22, E11=TheEqSymbOf S11,
E22=TheEqSymbOf S22, N11=TheNorSymbOf S11, N22=TheNorSymbOf S22,
AS11=AtomicFormulaSymbolsOf S11, AS22=AtomicFormulaSymbolsOf S22;
reconsider I22=I2 as Element of II22;
reconsider I11=I22|O11 as Element of II11 by FOMODEL2:2;
reconsider D11=S11-rules as isotone Correct 2-ranked RuleSet of S11;
dom a11=AS11 by FUNCT_2:def 1; then
A3: O11 c= dom a11 by FOMODEL1:1;
A4: now
let y be object; assume
A5: y in X2; then reconsider Y={y} as Subset of X2 by ZFMISC_1:31;
reconsider phi2=y as wff string of S22 by TARSKI:def 3, A5;
SymbolsOf Y=rng phi2 & SymbolsOf Y c= SymbolsOf X2 by FOMODEL0:45, 46;
then rng phi2 c= SymbolsOf X2 null L; then rng phi2 c= SS1 by XBOOLE_1:1;
then reconsider x=rng phi2/\O22 as Subset of O11 by A2, XBOOLE_1:26;
x c= dom a11 & a11 c= a22 by A3, FOMODEL1:def 41, XBOOLE_1:1; then
A6: a11|x=a22|x by GRFUNC_1:27; dom I11=O11 by PARTFUN1:def 2;
then I22|(rng phi2/\O22) = I11|(rng phi2/\O22) &
a22|(rng phi2/\O22) = a11|(rng phi2/\O22) & E11=E22 & N11=N22
by FOMODEL1:def 41, A6, GRFUNC_1:27; then
consider phi1 being wff string of S11 such that
A7: phi2=phi1 by FOMODEL3:16; thus y in FF11 by FOMODEL2:16, A7;
end;
now
let phi1 be wff string of S11;
O11 c= AS11 & dom a11=AS11 by FOMODEL1:1, FUNCT_2:def 1; then
N11 = N22 & E11 = E22 & I11|O11 = I22|O11 & a11|O11 = a22|O11
by GRFUNC_1:27, FOMODEL1:def 41;
then consider phi2 being wff string of S22 such that
A8: phi2=phi1 & I22-TruthEval phi2=I11-TruthEval phi1 by FOMODEL3:12;
assume phi1 in X2;
hence 1=I11-TruthEval phi1 by A8, A1;
end; then
X2 is D11-consistent by Lm53, FOMODEL2:def 42;
then consider U1 being countable non empty set,
I1 being Element of U1-InterpretersOf S11 such that
A9: X2 is I1-satisfied by Th19, A4, TARSKI:def 3;
set II=U1-InterpretersOf S22, I3=the Element of II;
reconsider
IT=(I3 +* I1)|O22 as Element of II by FOMODEL2:2; O11\O22 = {}; then
reconsider O111=O11 as non empty Subset of O22 by XBOOLE_1:37;
A10: IT|O11 =(I3+*I1)|(O111 null O22) by RELAT_1:71 .= I3|O11 +*(I1|O11)
by FUNCT_4:71 .= I1 null O11 .= I1|O11;
A11: N11=N22 & E11=E22 & a11|O11 = a22|O11 by A3, GRFUNC_1:27, FOMODEL1:def 41;
reconsider ITT=IT as Element of U1-InterpretersOf S2; take U1, ITT;
now
let phi be wff string of S22; assume
A12: phi in X2; then phi in FF11 by A4; then
reconsider phi1=phi as wff string of S11;
consider phi2 being wff string of S22 such that
A13: phi1=phi2 & I1-TruthEval phi1=IT-TruthEval phi2 by A10, A11, FOMODEL3:12;
thus 1=IT-TruthEval phi by A13, A12, A9;
end;
hence thesis;
end;
