set R = R5 S;
set Q = S -sequents ;

now

let Seqts, Seqts2 be Subset of (S -sequents); :: thesis: ( Seqts c= Seqts2 implies (R5 S) . Seqts c= (R5 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume B0: Seqts c= Seqts2 ; :: thesis: (R5 S) . Seqts c= (R5 S) . Seqts2

now

let x be set ; :: thesis: ( x in (R5 S) . Seqts implies x in (R5 S) . Seqts2 )
assume CC0: x in (R5 S) . Seqts ; :: thesis: x in (R5 S) . Seqts2
reconsider seqt = x as Element of S -sequents by CC0;
[Seqts,seqt] in P5 S by CC0, Lm1e;
then seqt Rule5 Seqts by DefP5;
then consider v1, v2 being literal Element of S, z being set , p being FinSequence such that
C1: ( seqt `1 = z \/ {(<*v1*> ^ p)} & not v2 is (z \/ {p}) \/ {(seqt `2)} -occurring & [(z \/ {((v1 SubstWith v2) . p)}),(seqt `2)] in Seqts ) by Def5;
seqt Rule5 Seqts2 by Def5, B0, C1;
then [Seqts2,seqt] in P5 S by DefP5;
hence x in (R5 S) . Seqts2 by Th3; :: thesis: verum

end;

hence (R5 S) . Seqts c= (R5 S) . Seqts2 by TARSKI:def 3; :: thesis: verum

end;

hence for b₁ being Rule of S st b₁ = R5 S holds
b₁ is isotone by DefMonotonic1; :: thesis: verum