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 being Function, a,b,A being set st a in A holds f,f+*(a,b)
  equal_outside A
proof
  let f be Function, a,b,A be set;
  per cases;
  suppose
A1: a in dom f;
    assume a in A;
    then {a} c= A by ZFMISC_1:31;
    then
A2: dom(a.-->b) c= A;
    f+*(a,b) = f+*(a.-->b) by A1,Def2;
    hence thesis by A2,Th28;
  end;
  suppose
    not a in dom f;
    then f+*(a,b) = f by Def2;
    hence thesis;
  end;
end;
