reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem
  for f being one-to-one PartFunc of C,D st d in rng f holds d = f/.(f"
  /.d) & d = (f*(f"))/.d
proof
  let f be one-to-one PartFunc of C,D;
  assume
A1: d in rng f;
  then d = ((f*f") qua Function).d & d in dom (f*f") by FUNCT_1:35,37;
  hence thesis by A1,Th11,PARTFUN1:def 6;
end;
