reserve x,x1,x2,y,y9,y1,y2,z,z1,z2 for object,P,X,X1,X2,Y,Y1,Y2,V,Z for set;

theorem
  for f1,f2 being PartFunc of X,{y} st dom f1 = dom f2 holds f1 = f2
proof
  let f1,f2 be PartFunc of X,{y} such that
A1: dom f1 = dom f2;
  for x being object holds x in dom f1 implies f1.x = f2.x
  proof let x be object;
    assume
A2: x in dom f1;
    then f1.x = y by Th20;
    hence thesis by A1,A2,Th20;
  end;
  hence thesis by A1;
end;
