reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;
reserve m,n,k for Nat, R for Relation;
reserve i,j for Nat;
reserve F for Function,
  e,x,y,z for object;

theorem
  for f be Function,d,r be set st d in dom f holds dom f=dom (f+*(d.-->r
  ))
proof
  let f be Function,d,r be set;
  assume
A1: d in dom f;
  thus dom f=dom (f+*(d,r)) by Th29
    .=dom (f+*(d.-->r)) by A1,Def2;
end;
