set R = R7 S;
set Q = S -sequents ;

now

let Seqts, Seqts2 be Subset of (S -sequents); :: thesis: ( Seqts c= Seqts2 implies (R7 S) . Seqts c= (R7 S) . Seqts2 )
set X = Seqts;
set Y = Seqts2;
assume B0: Seqts c= Seqts2 ; :: thesis: (R7 S) . Seqts c= (R7 S) . Seqts2

now

let x be set ; :: thesis: ( x in (R7 S) . Seqts implies x in (R7 S) . Seqts2 )
assume CC0: x in (R7 S) . Seqts ; :: thesis: x in (R7 S) . Seqts2
reconsider seqt = x as Element of S -sequents by CC0;
[Seqts,seqt] in P7 S by CC0, Lm1e;
then seqt Rule7 Seqts by DefP7;
then consider y being set , phi1, phi2 being wff string of S such that
C1: ( y in Seqts & y `1 = seqt `1 & y `2 = (<*(TheNorSymbOf S)*> ^ phi1) ^ phi2 & seqt `2 = (<*(TheNorSymbOf S)*> ^ phi2) ^ phi1 ) by Def7;
seqt Rule7 Seqts2 by Def7, C1, B0;
then [Seqts2,seqt] in P7 S by DefP7;
hence x in (R7 S) . Seqts2 by Th3; :: thesis: verum

end;

hence (R7 S) . Seqts c= (R7 S) . Seqts2 by TARSKI:def 3; :: thesis: verum

end;

hence for b₁ being Rule of S st b₁ = R7 S holds
b₁ is isotone by DefMonotonic1; :: thesis: verum