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;

theorem Th14:
(for R being Rule of S st R in D holds R is Correct) implies D is Correct
proof
set Q=S-sequents, O=OneStep D; {} null S is S-correct; then
reconsider e={} null Q as S-correct Subset of Q;
reconsider RO=rng O as Subset of bool Q by RELAT_1:def 19;
assume
A1: for R being Rule of S st R in D holds R is Correct;
defpred P[Nat] means for X being S-correct Subset of Q holds
($1,D)-derivables.X is S-correct;
A2: P[0]
proof
set f=(0,D)-derivables;
A3: f = id field O by FUNCT_7:68 .= id (bool Q\/RO) by FUNCT_2:def 1 .=
id(bool Q); let X be S-correct Subset of Q; thus thesis by A3;
end;
A4: for n st P[n] holds P[n+1]
proof
let n; assume
A5: P[n]; let X be S-correct Subset of Q;
set DM=(n+1,D)-derivables, Dm=(n,D)-derivables;
A6: dom Dm=bool Q by FUNCT_2:def 1;
reconsider oldSeqs=Dm.X as S-correct Subset of Q by A5;
A7: DM=O*Dm by FUNCT_7:71;
now
let U; set II=U-InterpretersOf S; let I be Element of II;
let H be I-satisfied set; let phi;
assume
A8: [H,phi] in DM.X;
set Fam={R.:{oldSeqs} where R is Subset of [:bool Q, bool Q:]: R in D};
DM.X=O.oldSeqs by A6, A7, FUNCT_1:13 .=
union union Fam by Lm5; then consider x such that
A9: [H,phi] in x & x in union Fam by A8, TARSKI:def 4; consider y such that
A10: x in y & y in Fam by A9, TARSKI:def 4; consider R being Subset of
[:bool Q, bool Q:] such that
A11: y=R.:{oldSeqs} & R in D by A10; reconsider RR=R as Correct Rule of S
by A1, A11; reconsider newSeqs=RR.oldSeqs as S-correct Subset of Q
by Def68;
dom RR=bool Q by FUNCT_2:def 1; then
y=Im(R,oldSeqs) &
Im(RR,oldSeqs)= {RR.oldSeqs} by FUNCT_1:59, A11; then
[H,phi] in newSeqs by A9, TARSKI:def 1, A10;
hence I-TruthEval phi = 1 by FOMODEL2:def 44;
end; hence thesis;
end;
A12: for n holds P[n] from NAT_1:sch 2(A2, A4);
now
let phi; let X; assume phi is (X,D)-provable; then
consider H being set, m such that
A13: H c= X & [H,phi] is (m,{},D)-derivable;
reconsider HH=H as Subset of X by A13;
reconsider seqt=[H,phi] as Element of Q by Def2, A13;
reconsider okSeqs=(m,D)-derivables.e as S-correct Subset of Q by A12;
hereby
let U; set II=U-InterpretersOf S; let I be Element of II; assume
X is I-satisfied; then reconsider XX=X as I-satisfied set;
reconsider HHH=HH as I-satisfied Subset of XX;
[HHH,phi] in okSeqs by A13;
hence I-TruthEval phi=1 by FOMODEL2:def 44;
end;
end; hence D is Correct;
end;
