reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;

theorem
  TotFuncs <:{},X,Y:> = Funcs(X,Y)
proof
  per cases;
  suppose
A1: Y = {} & X <> {};
    then TotFuncs <:{},X,Y:> = {};
    hence thesis by A1;
  end;
  suppose
A2: Y = {} implies X = {};
    for g being object holds g in TotFuncs <:{},X,Y:> iff g in Funcs(X,Y)
    proof
      let g be object;
      thus g in TotFuncs <:{},X,Y:> implies g in Funcs(X,Y)
      proof
        assume g in TotFuncs <:{},X,Y:>;
        then g is Function of X,Y by Th81;
        hence thesis by A2,Th8;
      end;
      assume
A3:   g in Funcs(X,Y);
      then reconsider g9 = g as PartFunc of X,Y by Th65;
A4:   <:{},X,Y:> tolerates g9 by PARTFUN1:60;
      g is Function of X,Y by A3,Th65;
      hence thesis by A2,A4,PARTFUN1:def 5;
    end;
    hence thesis by TARSKI:2;
  end;
end;
