set R = R#9 S;
set Q = S -sequents ;

now :: thesis: for Seqts, Seqts2 being Subset of (S -sequents) st Seqts c= Seqts2 holds
(R#9 S) . Seqts c= (R#9 S) . Seqts2

let Seqts, Seqts2 be Subset of (S -sequents); :: thesis: ( Seqts c= Seqts2 implies (R#9 S) . Seqts c= (R#9 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume A1: Seqts c= Seqts2 ; :: thesis: (R#9 S) . Seqts c= (R#9 S) . Seqts2

now :: thesis: for x being object st x in (R#9 S) . Seqts holds
x in (R#9 S) . Seqts2

let x be object ; :: thesis: ( x in (R#9 S) . Seqts implies x in (R#9 S) . Seqts2 )
assume A2: x in (R#9 S) . Seqts ; :: thesis: x in (R#9 S) . Seqts2
then A3: ( x in S -sequents & [Seqts,x] in P#9 S ) by Lm30;
reconsider seqt = x as Element of S -sequents by A2;
seqt Rule9 Seqts by A3, Def46;
then consider y being set , phi being wff string of S such that
A4: ( y in Seqts & seqt `2 = phi & y `1 = seqt `1 & y `2 = xnot (xnot phi) ) ;
seqt Rule9 Seqts2 by A4, A1;
then [Seqts2,seqt] in P#9 S by Def46;
hence x in (R#9 S) . Seqts2 by Th3; :: thesis: verum

end;

hence (R#9 S) . Seqts c= (R#9 S) . Seqts2 ; :: thesis: verum

end;

hence for b₁ being Rule of S st b₁ = R#9 S holds
b₁ is isotone ; :: thesis: verum