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 Th2:
  f is_one-to-one_at x iff x in dom f & f " {f.x} = {x}
proof
  thus f is_one-to-one_at x implies x in dom f & f " {f.x} = {x}
  proof
    assume
A1: f is_one-to-one_at x;
    hence
A2: x in dom f by Th1;
    f"Im(f,x) = {x} by A1;
    hence thesis by A2,FUNCT_1:59;
  end;
  assume x in dom f & f " {f.x} = {x};
  hence f"Im(f,x) = {x} by FUNCT_1:59;
end;
