now :: thesis: for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 holds
(R#0 S) . Seqts1 c= (R#0 S) . Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); :: thesis: ( Seqts1 c= Seqts2 implies (R#0 S) . Seqts1 c= (R#0 S) . Seqts2 )
set X = Seqts1;
set Y = Seqts2;
assume Seqts1 c= Seqts2 ; :: thesis: (R#0 S) . Seqts1 c= (R#0 S) . Seqts2
set R = R#0 S;
set Q = S -sequents ;
now :: thesis: for x being object st x in (R#0 S) . Seqts1 holds
x in (R#0 S) . Seqts2
let x be object ; :: thesis: ( x in (R#0 S) . Seqts1 implies x in (R#0 S) . Seqts2 )
assume A1: x in (R#0 S) . Seqts1 ; :: thesis: x in (R#0 S) . Seqts2
reconsider seqt = x as Element of S -sequents by A1;
[Seqts1,seqt] in P#0 S by A1, Lm30;
then seqt Rule0 Seqts1 by Def34;
then seqt `2 in seqt `1 ;
then seqt Rule0 Seqts2 ;
then [Seqts2,seqt] in P#0 S by Def34;
hence x in (R#0 S) . Seqts2 by Lm27; :: thesis: verum
end;
hence (R#0 S) . Seqts1 c= (R#0 S) . Seqts2 ; :: thesis: verum
end;
hence for b1 being Rule of S st b1 = R#0 S holds
b1 is isotone ; :: thesis: verum