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
  for f1 being Function of X,Y1 for f2 being Function of X,Y2 holds <:f1
  ,f2:> = [:f1,f2:]*(delta X)
proof
  let f1 be Function of X,Y1;
  let f2 be Function of X,Y2;
  per cases;
  suppose
    Y1 = {} or Y2 = {};
    then
A1: dom f1 = {} or dom f2 = {};
A2: dom[:f1,f2:] = [:dom f1, dom f2:] by Def8
      .= {} by A1,ZFMISC_1:90;
    dom<:f1,f2:> = dom f1 /\ dom f2 by Def7
      .= {} by A1;
    hence <:f1,f2:> = {}*delta X .= [:f1,f2:]*delta X by A2,RELAT_1:41;
  end;
  suppose
    Y1 <> {} & Y2 <> {};
    then dom f1 = X & dom f2 = X by FUNCT_2:def 1;
    hence thesis by Th68;
  end;
end;
