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 Th9:
  f is_one-to-one_at x iff x in dom f & f just_once_values f.x
proof
  thus f is_one-to-one_at x implies x in dom f & f just_once_values f.x
  proof
    assume
A1: f is_one-to-one_at x;
    hence x in dom f by Th1;
    {x} = f " {f.x} by A1,Th2;
    hence thesis by Th6;
  end;
  assume that
A2: x in dom f and
A3: f just_once_values f.x;
  consider z such that
A4: f " {f.x} = {z} by A3,Th6;
  f.x in {f.x} by TARSKI:def 1;
  then x in {z} by A2,A4,FUNCT_1:def 7;
  then x = z by TARSKI:def 1;
  hence thesis by A2,A4,Th2;
end;
