let S be Language; :: thesis: R1 S is Correct

now

set f = R1 S;
set R = P1 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 (R1 S) . X is S -correct )
assume B0: X is S -correct ; :: thesis: (R1 S) . X is S -correct

now

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 (R1 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 (R1 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 (R1 S) . X holds
I -TruthEval psi = 1
let psi be wff string of S; :: thesis: ( [x,psi] in (R1 S) . X implies I -TruthEval psi = 1 )
set s = [x,psi];
set TE = I -TermEval ;
set d = U -deltaInterpreter ;
CC0: ( ([x,psi] `1) \+\ x = {} & ([x,psi] `2) \+\ psi = {} ) ;
then C0: ( dom (P1 S) c= bool (S -sequents) & [x,psi] `1 = x & [x,psi] `2 = psi ) by FOMODEL0:29;
assume DD1: [x,psi] in (R1 S) . X ; :: thesis: I -TruthEval psi = 1
then D1: ( [x,psi] in S -sequents & [X,[x,psi]] in P1 S ) by Lm1e;
then X in dom (P1 S) by RELAT_1:def 4;
then reconsider Seqts = X as S -correct Subset of (S -sequents) by B0;
reconsider seqt = [x,psi] as Element of S -sequents by DD1, Lm1e;
seqt Rule1 Seqts by D1, DefP1;
then consider y being set such that
D2: ( y in Seqts & y `1 c= seqt `1 & seqt `2 = y `2 ) by Def1;
reconsider H = y `1 as Subset of x by CC0, D2, FOMODEL0:29;
[H,psi] in Seqts by D2, C0, MCART_1:21;
hence I -TruthEval psi = 1 by FOMODEL2:def 44; :: thesis: verum

end;

hence (R1 S) . X is S -correct by FOMODEL2:def 44; :: thesis: verum

end;

hence R1 S is Correct by RuleCorr; :: thesis: verum