reserve k,m,n for Nat, kk,mm,nn for Element of NAT,
 U,U1,U2 for non empty set,
 A,B,X,Y,Z, x,x1,x2,y,z for set,
 S for Language, s, s1, s2 for Element of S,
f,g for Function, w for string of S, tt,tt1,tt2 for Element of AllTermsOf S,
psi,psi1,psi2,phi,phi1,phi2 for wff string of S, u,u1,u2 for Element of U,
Phi,Phi1,Phi2 for Subset of AllFormulasOf S, t,t1,t2,t3 for termal string of
S,
r for relational Element of S, a for ofAtomicFormula Element of S,
l, l1, l2 for literal Element of S, p for FinSequence,
m1, n1 for non zero Nat, S1, S2 for Language;
reserve D,D1,D2,D3 for RuleSet of S, R for Rule of S,
Seqts,Seqts1,Seqts2 for Subset of S-sequents,
seqt,seqt1,seqt2 for Element of S-sequents,
SQ,SQ1,SQ2 for S-sequents-like set, Sq,Sq1,Sq2 for S-sequent-like object;
reserve H,H1,H2,H3 for S-premises-like set;
reserve M,K,K1,K2 for isotone RuleSet of S;

theorem
{y} is (SQ,D)-derivable implies ex mm st y is (mm,SQ,D)-derivable
proof
set Q=S-sequents; reconsider Seqts=SQ as Element of bool Q
by Def3; {y} is (Seqts,D)-derivable implies ex mm st
y is (mm,Seqts,D)-derivable by Lm8; hence thesis;
end;
