let S be Language; :: thesis: R#1 S is Correct

now :: thesis: for X being set st X is S -correct holds
(R#1 S) . X is S -correct

set f = R#1 S;
set R = P#1 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#1 S) . X is S -correct )
assume A1: X is S -correct ; :: thesis: (R#1 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#1 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#1 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#1 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#1 S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; :: thesis: ( [x,psi] in (R#1 S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
set TE = I -TermEval ;
set d = U -deltaInterpreter ;
assume A4: [x,psi] in (R#1 S) . X ; :: thesis: I -TruthEval psi = 1
then A5: ( [x,psi] in S -sequents & [X,[x,psi]] in P#1 S ) by Lm30;
then X in dom (P#1 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 A4, Lm30;
seqt Rule1 Seqts by A5, Def35;
then consider y being set such that
A6: ( y in Seqts & y `1 c= seqt `1 & seqt `2 = y `2 ) ;
reconsider H = y `1 as Subset of x by A6;
[H,psi] in Seqts by A6, MCART_1:21;
hence I -TruthEval psi = 1 by FOMODEL2:def 44; :: thesis: verum

end;

hence (R#1 S) . X is S -correct ; :: thesis: verum

end;

hence R#1 S is Correct ; :: thesis: verum