set R = R1 S;
set Q = S -sequents ;

now

let Seqts, Seqts2 be Subset of (S -sequents); :: thesis: ( Seqts c= Seqts2 implies (R1 S) . Seqts c= (R1 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume B0: Seqts c= Seqts2 ; :: thesis: (R1 S) . Seqts c= (R1 S) . Seqts2

now

let x be set ; :: thesis: ( x in (R1 S) . Seqts implies x in (R1 S) . Seqts2 )
assume CC0: x in (R1 S) . Seqts ; :: thesis: x in (R1 S) . Seqts2
reconsider seqt = x as Element of S -sequents by CC0;
[Seqts,seqt] in P1 S by CC0, Lm1e;
then seqt Rule1 Seqts by DefP1;
then consider y being set such that
C1: ( y in Seqts & y `1 c= seqt `1 & seqt `2 = y `2 ) by Def1;
seqt Rule1 Seqts2 by Def1, C1, B0;
then [Seqts2,seqt] in P1 S by DefP1;
hence x in (R1 S) . Seqts2 by Th3; :: thesis: verum

end;

hence (R1 S) . Seqts c= (R1 S) . Seqts2 by TARSKI:def 3; :: thesis: verum

end;

hence for b₁ being Rule of S st b₁ = R1 S holds
b₁ is isotone by DefMonotonic1; :: thesis: verum