set R = R4 S;
set Q = S -sequents ;

now

let Seqts, Seqts2 be Subset of (S -sequents); :: thesis: ( Seqts c= Seqts2 implies (R4 S) . Seqts c= (R4 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume Seqts c= Seqts2 ; :: thesis: (R4 S) . Seqts c= (R4 S) . Seqts2

now

let x be set ; :: thesis: ( x in (R4 S) . Seqts implies x in (R4 S) . Seqts2 )
assume CC0: x in (R4 S) . Seqts ; :: thesis: x in (R4 S) . Seqts2
reconsider seqt = x as Element of S -sequents by CC0;
[Seqts,seqt] in P4 S by CC0, Lm1e;
then seqt Rule4 Seqts by DefP4;
then consider l being literal Element of S, phi being wff string of S, t being termal string of S such that
C1: ( seqt `1 = {((l,t) SubstIn phi)} & seqt `2 = <*l*> ^ phi ) by Def4;
seqt Rule4 Seqts2 by C1, Def4;
then [Seqts2,seqt] in P4 S by DefP4;
hence x in (R4 S) . Seqts2 by Th3; :: thesis: verum

end;

hence (R4 S) . Seqts c= (R4 S) . Seqts2 by TARSKI:def 3; :: thesis: verum

end;

hence for b₁ being Rule of S st b₁ = R4 S holds
b₁ is isotone by DefMonotonic1; :: thesis: verum