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 Th91:
  for f being Function, a,b being object,
    A being set holds a in A or (f+*(a,b))
  |A = f|A
proof
  let f be Function, a,b be object, A be set;
  per cases;
  suppose
A1: a in dom f;
    assume not a in A;
    then {a} misses A by ZFMISC_1:50;
    then
A2: dom (a.-->b) misses A;
    thus (f+*(a,b))|A = (f +* (a.-->b))|A by A1,Def2
      .= f|A by A2,FUNCT_4:72;
  end;
  suppose
    not a in dom f;
    hence thesis by Def2;
  end;
end;
