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,f2 being Function of X,Y for g1,g2 being Function of X,Z st (Y
= {} implies X = {}) & (Z = {} implies X = {}) & <:f1,g1:> = <:f2,g2:> holds f1
  = f2 & g1 = g2
proof
  let f1,f2 be Function of X,Y;
  let g1,g2 be Function of X,Z;
  assume that
A1: Y = {} implies X = {} and
A2: Z = {} implies X = {};
A3: dom g1 = X & dom g2 = X by A2,FUNCT_2:def 1;
  dom f1 = X & dom f2 = X by A1,FUNCT_2:def 1;
  hence thesis by A3,Th54;
end;
