now

let Seqts1, Seqts2 be Subset of (S -sequents); :: thesis: ( Seqts1 c= Seqts2 implies (R3b S) . Seqts1 c= (R3b S) . Seqts2 )
set X = Seqts1;
set Y = Seqts2;
assume Seqts1 c= Seqts2 ; :: thesis: (R3b S) . Seqts1 c= (R3b S) . Seqts2
set R = R3b S;
set Q = S -sequents ;

now

let x be set ; :: thesis: ( x in (R3b S) . Seqts1 implies x in (R3b S) . Seqts2 )
assume CC0: x in (R3b S) . Seqts1 ; :: thesis: x in (R3b S) . Seqts2
reconsider seqt = x as Element of S -sequents by CC0;
[Seqts1,seqt] in P3b S by CC0, Lm1e;
then seqt Rule3b Seqts1 by DefP3b;
then ex t1, t2 being termal string of S st
( seqt `1 = {((<*(TheEqSymbOf S)*> ^ t1) ^ t2)} & seqt `2 = (<*(TheEqSymbOf S)*> ^ t2) ^ t1 ) by Def3b;
then seqt Rule3b Seqts2 by Def3b;
then [Seqts2,seqt] in P3b S by DefP3b;
hence x in (R3b S) . Seqts2 by Lm1; :: thesis: verum

end;

hence (R3b S) . Seqts1 c= (R3b S) . Seqts2 by TARSKI:def 3; :: thesis: verum

end;

hence for b₁ being Rule of S st b₁ = R3b S holds
b₁ is isotone by DefMonotonic1; :: thesis: verum