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;
