let S be Language; :: thesis: R#7 S is Correct

now :: thesis: for X being set st X is S -correct holds
(R#7 S) . X is S -correct

set f = R#7 S;
set R = P#7 S;
set Q = S -sequents ;
set E = TheEqSymbOf S;
set N = TheNorSymbOf S;
let X be set ; :: thesis: ( X is S -correct implies (R#7 S) . X is S -correct )
assume A1: X is S -correct ; :: thesis: (R#7 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#7 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#7 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#7 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#7 S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; :: thesis: ( [x,psi] in (R#7 S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
assume A3: [x,psi] in (R#7 S) . X ; :: thesis: I -TruthEval psi = 1
then A4: ( [x,psi] in S -sequents & [X,[x,psi]] in P#7 S ) by Lm30;
then X in dom (P#7 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 Rule7 Seqts by A4, Def44;
then consider y being set , phi1, phi2 being wff string of S such that
A5: ( y in Seqts & y `1 = seqt `1 & y `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1 ) ;
( psi = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1 & [x,((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2)] in Seqts ) by A5, MCART_1:21;
then I -TruthEval ((<*(TheNorSymbOf S)*> ^ phi1) ^ phi2) = 1 by FOMODEL2:def 44;
then ( I -TruthEval phi1 = 0 & I -TruthEval phi2 = 0 ) by FOMODEL2:19;
hence I -TruthEval psi = 1 by A5, FOMODEL2:19; :: thesis: verum

end;

hence (R#7 S) . X is S -correct ; :: thesis: verum

end;

hence R#7 S is Correct ; :: thesis: verum