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 ::#Goedel's completeness theorem
(X c= AllFormulasOf C & phi is X-implied) implies phi is X-provable
proof
reconsider S=C as Language; reconsider DD={R#9(S)} as RuleSet of S;
set FF=AllFormulasOf C, D=C-rules; assume X c= FF; then
reconsider Y=X as Subset of FF; assume phi is X-implied; then
reconsider phii=phi as X-implied wff string of C; set psi=xnot xnot phii;
psi is (Y,D)-provable by Lm77; then consider H being set, m such that
A1: H c= Y & [H, psi] is (m,{},D)-derivable;
reconsider seqt=[H, psi] as C-sequent-like object by A1;
A2: seqt`1 \+\ H={};
reconsider HH=H as S-premises-like set by A2;
reconsider HC=H as C-premises-like set by A2;
reconsider a=phi as wff string of S;
[HC, phi] null 1 is (1,{[HC, xnot (xnot phi)]}, {R#9(C)})-derivable;
then [HC, phi] is (m+1, {}, D\/{R#9(C)})-derivable by Lm22, A1; then
phi is (Y,D\/{R#9(C)})-provable by A1;
hence thesis;
end;
