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;
