now :: thesis: for Seqts1, Seqts2 being Subset of (S -sequents) st Seqts1 c= Seqts2 holds
(R#3e S) . Seqts1 c= (R#3e S) . Seqts2
let Seqts1, Seqts2 be Subset of (S -sequents); :: thesis: ( Seqts1 c= Seqts2 implies (R#3e S) . Seqts1 c= (R#3e S) . Seqts2 )
set X = Seqts1;
set Y = Seqts2;
assume Seqts1 c= Seqts2 ; :: thesis: (R#3e S) . Seqts1 c= (R#3e S) . Seqts2
set R = R#3e S;
set Q = S -sequents ;
now :: thesis: for x being object st x in (R#3e S) . Seqts1 holds
x in (R#3e S) . Seqts2
let x be object ; :: thesis: ( x in (R#3e S) . Seqts1 implies x in (R#3e S) . Seqts2 )
assume A5: x in (R#3e S) . Seqts1 ; :: thesis: x in (R#3e S) . Seqts2
reconsider seqt = x as Element of S -sequents by A5;
[Seqts1,seqt] in P#3e S by A5, Lm30;
then seqt Rule3e Seqts1 by Def40;
then ex s being relational Element of S ex T, U being |.(ar b1).| -element Element of (AllTermsOf S) * st
( seqt `1 = {(s -compound T)} \/ { ((<*(TheEqSymbOf S)*> ^ (TT . j)) ^ (UU . j)) where j is Element of Seg |.(ar s).|, TT, UU is Function of (Seg |.(ar s).|),(((AllSymbolsOf S) *) \ {{}}) : ( TT = T & UU = U ) } & seqt `2 = s -compound U ) ;
then seqt Rule3e Seqts2 ;
then [Seqts2,seqt] in P#3e S by Def40;
hence x in (R#3e S) . Seqts2 by Lm27; :: thesis: verum
end;
hence (R#3e S) . Seqts1 c= (R#3e S) . Seqts2 ; :: thesis: verum
end;
hence for b1 being Rule of S st b1 = R#3e S holds
b1 is isotone ; :: thesis: verum