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;
