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 Th74:
  for f being Function of X,Y for g being Function of V,Z holds [:
  f,g:] is Function of [:X,V:],[:Y,Z:]
proof
  let f be Function of X,Y;
  let g be Function of V,Z;
  per cases;
  suppose
A1: [:Y,Z:] = {} implies [:X,V:] = {};
    now
      per cases by A1;
      suppose
A2:     [:Y,Z:] <> {};
        rng f c= Y & rng g c= Z by RELAT_1:def 19;
        then [:rng f,rng g:] c= [:Y,Z:] by ZFMISC_1:96;
        then
A3:     rng [:f,g:] c= [:Y,Z:] by Th67;
        Z = {} implies V = {} by A2,ZFMISC_1:90;
        then
A4:     dom g = V by FUNCT_2:def 1;
        Y = {} implies X = {} by A2,ZFMISC_1:90;
        then dom f = X by FUNCT_2:def 1;
        then dom[:f,g:] = [:X,V:] by A4,Def8;
        hence thesis by A3,FUNCT_2:2;
      end;
      suppose
A5:     [:X,V:] = {};
        then X = {} or V = {};
        then dom f = {} or dom g = {};
        then [:dom f,dom g:] = {} by ZFMISC_1:90;
        then
A6:     dom[:f,g:] = [:X,V:] by A5,Def8;
        rng f c= Y & rng g c= Z by RELAT_1:def 19;
        then [:rng f,rng g:] c= [:Y,Z:] by ZFMISC_1:96;
        then rng[:f,g:] c= [:Y,Z:] by Th67;
        hence thesis by A6,FUNCT_2:2;
      end;
    end;
    hence thesis;
  end;
  suppose
A7: [:Y,Z:] = {} & [:X,V:] <> {};
    then Y = {} or Z = {};
    then f = {} or g = {};
    then [:dom f,dom g:] = {} by ZFMISC_1:90;
    then
A8: dom [:f,g:] = {} by Def8;
    then rng[:f,g:] = {} & dom [:f,g:] c= [:X,V:] by RELAT_1:42;
    then reconsider R = [:f,g:] as Relation of [:X,V:],[:Y,Z:] by RELSET_1:4
,XBOOLE_1:2;
    [:f,g:] = {} by A8;
    then R is quasi_total by A7,FUNCT_2:def 1;
    hence thesis;
  end;
end;
