reserve V, C for set;
reserve A, B, D for Element of Fin PFuncs (V, C);

theorem Th4:
  B = { {} } implies A ^ B = A
proof
A1: { s \/ t where s,t is Element of PFuncs (V,C) : s in A & t in { {} } & s
  tolerates t } c= A
  proof
    let a be object;
    assume a in { s \/ t where s,t is Element of PFuncs (V,C) : s in A & t in
    { {} } & s tolerates t };
    then consider s9, t9 being Element of PFuncs (V,C) such that
A2: a = s9 \/ t9 & s9 in A and
A3: t9 in { {} } and
    s9 tolerates t9;
    t9 = {} by A3,TARSKI:def 1;
    hence thesis by A2;
  end;
A4: A c= { s \/ t where s,t is Element of PFuncs (V,C) : s in A & t in { {}
  } & s tolerates t }
  proof
    {} is PartFunc of V, C by RELSET_1:12;
    then reconsider b = {} as Element of PFuncs (V,C) by PARTFUN1:45;
    let a be object;
    assume
A5: a in A;
    A c= PFuncs (V,C) by FINSUB_1:def 5;
    then reconsider a9 = a as Element of PFuncs (V,C) by A5;
A6: b in { {} } by TARSKI:def 1;
    a = a9 \/ b & a9 tolerates b by PARTFUN1:59;
    hence thesis by A5,A6;
  end;
  assume B = { {} };
  hence thesis by A1,A4;
end;
