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 c in dom f holds c = f"/.(f
  /.c) & c = (f"*f)/.c
proof
  let f be one-to-one PartFunc of C,D;
  assume
A1: c in dom f;
  hence
A2: c = f"/.(f/.c) by Th11;
  f" = f";
  then f/.c in rng f by A1,Th11;
  then f/.c in dom (f") by FUNCT_1:33;
  hence thesis by A1,A2,Th4;
end;
