reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;
reserve i for Nat;
reserve m for Nat,
        D for non empty set;

theorem Th62:
  for F being non empty initial NAT-defined Function holds 0 in dom F
proof
 let F be non empty initial NAT-defined Function;
  consider x being object such that
A1: x in dom F by XBOOLE_0:def 1;
  dom F c= NAT by RELAT_1:def 18;
  then reconsider x as Element of NAT by A1;
  x = 0 or 0 < x;
  hence 0 in dom F by A1,Def12;
end;
