reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem Th45:
  Funcs({},f) = dom f --> {{}}
proof
A1: now
    let x be object;
    assume x in dom f;
    hence Funcs({},f).x = Funcs({} qua set,f.x) by Def8
      .= {{}} by FUNCT_5:57;
  end;
  dom Funcs({},f) = dom f by Def8;
  hence thesis by A1,FUNCOP_1:11;
end;
