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;

theorem
  for f,g being Function, x,y being set st g c= f & not x in dom g holds
  g c= f+*(x,y)
proof
  let f,g be Function, x,y be set such that
A1: g c= f and
A2: not x in dom g;
A3: now
    let z be object;
    assume
A4: z in dom g;
    hence g.z = f.z by A1,GRFUNC_1:2
      .= (f+*(x,y)).z by A2,A4,Th31;
  end;
  dom g c= dom f by A1,GRFUNC_1:2;
  then dom g c= dom(f+*(x,y)) by Th29;
  hence thesis by A3,GRFUNC_1:2;
end;
