reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;

theorem Th1:
  f is_one-to-one_at x implies x in dom f
proof
  assume f is_one-to-one_at x;
  then f"Im(f,x) = {x};
  then x in f"Im(f,x) by TARSKI:def 1;
  hence thesis by FUNCT_1:def 7;
end;
