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 Th114:
  for A, B, a, b being set, f being Function of A,B st b in B
  holds f +* (a,b) is Function of A,B
proof
  let A, B, a, b be set, f be Function of A,B such that
A1: b in B;
A2: dom f = A by A1,FUNCT_2:def 1;
  per cases;
  suppose
    not a in A;
    hence thesis by A2,Def2;
  end;
  suppose
A3: a in A;
    set g = a .--> b;
    set f1 = f +* g;
    rng g = {b} by FUNCOP_1:8;
    then rng g c= B by A1,ZFMISC_1:31;
    then
A4: rng f1 c= rng f \/ rng g & rng f \/ rng g c= B \/ B by FUNCT_4:17
,XBOOLE_1:13;
A5: {a} c= A by A3,ZFMISC_1:31;
    dom f1 = A \/ dom g by A2,FUNCT_4:def 1
      .= A \/ {a}
      .= A by A5,XBOOLE_1:12;
    then
A6: f +* g is Function of A,B by A4,FUNCT_2:2,XBOOLE_1:1;
    a in dom f by A1,A3,FUNCT_2:def 1;
    hence thesis by A6,Def2;
  end;
end;
