set R = R#8 S;
set Q = S -sequents ;

now :: thesis: for Seqts, Seqts2 being Subset of (S -sequents) st Seqts c= Seqts2 holds
(R#8 S) . Seqts c= (R#8 S) . Seqts2

let Seqts, Seqts2 be Subset of (S -sequents); :: thesis: ( Seqts c= Seqts2 implies (R#8 S) . Seqts c= (R#8 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume A1: Seqts c= Seqts2 ; :: thesis: (R#8 S) . Seqts c= (R#8 S) . Seqts2

now :: thesis: for x being object st x in (R#8 S) . Seqts holds
x in (R#8 S) . Seqts2

let x be object ; :: thesis: ( x in (R#8 S) . Seqts implies x in (R#8 S) . Seqts2 )
assume A2: x in (R#8 S) . Seqts ; :: thesis: x in (R#8 S) . Seqts2
reconsider seqt = x as Element of S -sequents by A2;
[Seqts,seqt] in P#8 S by A2, Lm30;
then seqt Rule8 Seqts by Def45;
then consider y1, y2 being set , phi, phi1, phi2 being wff string of S such that
A3: ( y1 in Seqts & y2 in Seqts & y1 `1 = y2 `1 & y1 `2 = phi1 & y2 `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & {phi} \/ (seqt `1) = y1 `1 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi) ^ phi ) ;
seqt Rule8 Seqts2 by A3, A1;
then [Seqts2,seqt] in P#8 S by Def45;
hence x in (R#8 S) . Seqts2 by Th3; :: thesis: verum

end;

hence (R#8 S) . Seqts c= (R#8 S) . Seqts2 ; :: thesis: verum

end;

hence for b₁ being Rule of S st b₁ = R#8 S holds
b₁ is isotone ; :: thesis: verum