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 Th61:
  for f being Function of X,Y for g being Function of X,Z st (Y =
{} implies X = {}) & (Z = {} implies X = {}) holds pr1(Y,Z)*<:f,g:> = f & pr2(Y
  ,Z)*<:f,g:> = g
proof
  let f be Function of X,Y;
  let g be Function of X,Z;
  assume ( Y = {} implies X = {})&( Z = {} implies X = {});
  then
A1: dom f = X & dom g = X by FUNCT_2:def 1;
  rng f c= Y & rng g c= Z by RELAT_1:def 19;
  hence thesis by A1,Th52;
end;
