reserve V for Universe,
  a,b,x,y,z,x9,y9 for Element of V,
  X for Subclass of V,
  o,p,q,r,s,t,u,a1,a2,a3,A,B,C,D for set,
  K,L,M for Ordinal,
  n for Element of omega,
  fs for finite Subset of omega,
  e,g,h for Function,
  E for non empty set,
  f for Function of VAR,E,
  k,k1 for Element of NAT,
  v1,v2,v3 for Element of VAR,
  H,H9 for ZF-formula;

theorem Th14:
  X is closed_wrt_A1-A7 & A c= X & y in Funcs(fs,A) implies y in X
proof
  assume that
A1: X is closed_wrt_A1-A7 and
A2: A c= X and
A3: y in Funcs(fs,A);
  consider g such that
A4: y=g and
A5: dom g=fs and
A6: rng g c= A by A3,FUNCT_2:def 2;
A7: now
    let o;
    assume
A8: o in y;
    then consider p,q being object such that
A9: o=[p,q] by A4,RELAT_1:def 1;
A10: p in dom g by A4,A8,A9,FUNCT_1:1;
    q=g.p by A4,A8,A9,FUNCT_1:1;
    then q in rng g by A10,FUNCT_1:def 3;
    then
A11: q in A by A6;
A12: omega c= X by A1,Th7;
    p in omega by A5,A10;
    hence o in X by A1,A2,A9,A12,A11,Th6;
  end;
  rng g is finite by A5,FINSET_1:8;
  then [:dom g,rng g:] is finite by A5;
  then y is finite by A4,FINSET_1:1,RELAT_1:7;
  hence thesis by A1,A7,Th13;
end;
