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;
