let S be Language; :: thesis: R#0 S is Correct

now :: thesis: for X being set st X is S -correct holds
(R#0 S) . X is S -correct

set f = R#0 S;
set R = P#0 S;
set Q = S -sequents ;
let X be set ; :: thesis: ( X is S -correct implies (R#0 S) . X is S -correct )
assume X is S -correct ; :: thesis: (R#0 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 phi being wff string of S st [x,phi] in (R#0 S) . X holds
I -TruthEval phi = 1

let U be non empty set ; :: thesis: for I being Element of U -InterpretersOf S
for x being b₁ -satisfied set
for phi being wff string of S st [x,phi] in (R#0 S) . X holds
I -TruthEval phi = 1
set II = U -InterpretersOf S;
let I be Element of U -InterpretersOf S; :: thesis: for x being I -satisfied set
for phi being wff string of S st [x,phi] in (R#0 S) . X holds
I -TruthEval phi = 1
let x be I -satisfied set ; :: thesis: for phi being wff string of S st [x,phi] in (R#0 S) . X holds
I -TruthEval phi = 1
let phi be wff string of S; :: thesis: ( [x,phi] in (R#0 S) . X implies I -TruthEval phi = 1 )
set s = [x,phi];
assume A2: [x,phi] in (R#0 S) . X ; :: thesis: I -TruthEval phi = 1
then A3: ( [x,phi] in S -sequents & [X,[x,phi]] in P#0 S ) by Lm30;
then X in dom (P#0 S) by XTUPLE_0:def 12;
then reconsider XX = X as Subset of (S -sequents) ;
reconsider seqt = [x,phi] as Element of S -sequents by A2, Lm30;
seqt Rule0 XX by A3, Def34;
hence I -TruthEval phi = 1 by FOMODEL2:def 42; :: thesis: verum

end;

hence (R#0 S) . X is S -correct ; :: thesis: verum

end;

hence R#0 S is Correct ; :: thesis: verum