reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th58:
  for f being Function of X,Y for g being Function of X,Z st (Y =
{} implies X = {}) & (Z = {} implies X = {}) holds <:f,g:> is Function of X,[:Y
  ,Z:]
proof
  let f be Function of X,Y;
  let g be Function of X,Z;
  assume
A1: ( Y = {} implies X = {})&( Z = {} implies X = {});
  per cases;
  suppose
 [:Y,Z:] = {} implies X = {};
    rng f c= Y & rng g c= Z by RELAT_1:def 19;
    then
A2: [:rng f,rng g:] c= [:Y,Z:] by ZFMISC_1:96;
    dom f = X & dom g = X by A1,FUNCT_2:def 1;
    then
A3: dom<:f,g:> = X by Th50;
    rng <:f,g:> c= [:rng f,rng g:] by Th51;
    then rng<:f,g:> c= [:Y,Z:] by A2;
    hence thesis by A3,FUNCT_2:2;
  end;
  suppose
    [:Y,Z:] = {} & X <> {};
    hence thesis by A1;
  end;
end;
