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;

proof ::#Satisfiability theorem
let S be countable Language; set S1=S; let D be RuleSet of S1;
set FF1=AllFormulasOf S1; assume
A1: D is 2-ranked & D is isotone & D is Correct
& Z is D-consistent & Z c= FF1; then reconsider X=Z as Subset of FF1;
set S2=S1 addLettersNotIn X, O1=OwnSymbolsOf S1, O2=OwnSymbolsOf S2,
FF2=AllFormulasOf S2, SS1=AllSymbolsOf S1, SS2=AllSymbolsOf S2,
strings2=SS2*\{{}}, L2=LettersOf S2;
reconsider D1=D as 2-ranked Correct RuleSet of S1 by A1; O1\O2 ={}; then
reconsider O11=O1 as non empty Subset of O2 by XBOOLE_1:37;
reconsider D2=S2-rules as 2-ranked Correct isotone RuleSet of S2;
reconsider sub1=X/\strings2 as Subset of X;
reconsider sub2=SymbolsOf sub1 as Subset of SymbolsOf X by FOMODEL0:46;
reconsider inf=L2\SymbolsOf X as Subset of L2\sub2 by XBOOLE_1:34;
A2: L2\sub2 null inf is infinite;
now
let Y be finite Subset of X; reconsider YY=Y as functional set;
reconsider YYY=YY as functional Subset of FF1 by XBOOLE_1:1;
YY is finite & FF1 is countable & YY is D1-consistent & D1 is isotone
by A1; then consider U being non empty countable set such that
A3: ex I1 being Element of U-InterpretersOf S1 st YY is I1-satisfied
by Lm76; set II1=U-InterpretersOf S1, II2=U-InterpretersOf S2,
I02=the (S2,U)-interpreter-like Function;
consider I1 being Element of II1 such that
A4: YYY is I1-satisfied by A3;
reconsider I2 = (I02 +* I1)|O2 as Element of II2 by FOMODEL2:2;
I2|O1 = (I02 +* I1)|(O11 null O2) by RELAT_1:71
.= I02|O1 +* (I1|O1) by FUNCT_4:71 .= I1|O1; then
YYY is I2-satisfied by A4, FOMODEL3:17; hence Y is D2-consistent by Lm53;
end; then X is D2-consistent by Lm51; then consider U being
non empty countable set, I2 being Element of U-InterpretersOf S2 such that
A5: X is I2-satisfied by A2, Lm75;
set II1=U-InterpretersOf S1, II2=U-InterpretersOf S2;
take U; reconsider I1=I2|O1 as Element of II1 by FOMODEL2:2;
take I1; I1|O1=I2|O1 null O1;
hence thesis by A5, FOMODEL3:17;
end;
