let S be Language; :: thesis: R#3a S is Correct

now :: thesis: for X being set st X is S -correct holds
(R#3a S) . X is S -correct

set f = R#3a S;
set R = P#3a S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
set FF = AllFormulasOf S;
set TT = AllTermsOf S;
set SS = AllSymbolsOf S;
set F = S -firstChar ;
set C = S -multiCat ;
let X be set ; :: thesis: ( X is S -correct implies (R#3a S) . X is S -correct )
assume A1: X is S -correct ; :: thesis: (R#3a S) . X is S -correct

now :: thesis: for U being non empty set
for I being Element of U -InterpretersOf S
for x being b₂ -satisfied set
for psi being wff string of S st [x,psi] in (R#3a S) . X holds
I -TruthEval psi = 1

let U be non empty set ; :: thesis: for I being Element of U -InterpretersOf S
for x being b₁ -satisfied set
for psi being wff string of S st [x,psi] in (R#3a S) . X holds
I -TruthEval psi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; :: thesis: for x being I -satisfied set
for psi being wff string of S st [x,psi] in (R#3a S) . X holds
I -TruthEval psi = 1
let x be I -satisfied set ; :: thesis: for psi being wff string of S st [x,psi] in (R#3a S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; :: thesis: ( [x,psi] in (R#3a S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
set TE = I -TermEval ;
set d = U -deltaInterpreter ;
assume A3: [x,psi] in (R#3a S) . X ; :: thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#3a S ) by Lm30;
then X in dom (P#3a S) by XTUPLE_0:def 12;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by A1;
reconsider seqt = [x,psi] as Element of S -sequents by A3, Lm30;
seqt Rule3a Seqts by A4, Def37;
then consider t1, t2, t3 being termal string of S such that
A5: seqt = [{((<*(TheEqSymbOf S)*> ^ t1) ^ t2),((<*(TheEqSymbOf S)*> ^ t2) ^ t3)},((<*(TheEqSymbOf S)*> ^ t1) ^ t3)] ;
reconsider phi1 = (<*(TheEqSymbOf S)*> ^ t1) ^ t2, phi2 = (<*(TheEqSymbOf S)*> ^ t2) ^ t3, phi = (<*(TheEqSymbOf S)*> ^ t1) ^ t3 as 0wff string of S ;
A6: ( {phi1,phi2} \+\ (seqt `1) = {} & phi \+\ (seqt `2) = {} ) by A5;
then ( {phi1,phi2} is I -satisfied & phi = psi & {phi1} \ {phi1,phi2} = {} & {phi2} \ {phi1,phi2} = {} ) by FOMODEL0:29;
then ( I -TruthEval phi1 = 1 & I -TruthEval phi2 = 1 ) by ZFMISC_1:60;
then ( (I -TermEval) . t1 = (I -TermEval) . t2 & (I -TermEval) . t2 = (I -TermEval) . t3 ) by Lm54;
then I -TruthEval phi = 1 by Lm54;
hence I -TruthEval psi = 1 by A6, FOMODEL0:29; :: thesis: verum

end;

hence (R#3a S) . X is S -correct ; :: thesis: verum

end;

hence R#3a S is Correct ; :: thesis: verum