reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th6:
  for Y being non empty set for p being Function of X,Y st p is
  one-to-one for x1,x2 being Subset of X holds x1 <> x2 implies p.:x1 <> p.:x2
proof
  let Y be non empty set;
  let p be Function of X,Y such that
A1: p is one-to-one;
  let x1 be Subset of X;
  let x2 be Subset of X;
A2: X = dom p by FUNCT_2:def 1;
A3: not x1 c= x2 implies p.:x1 <> p.:x2
  proof
    assume not x1 c= x2;
    then consider a being object such that
A4: a in x1 and
A5: not a in x2;
    not p.a in p.:x2
    proof
      assume p.a in p.:x2;
      then ex b being object st b in dom p & b in x2 & p.a = p.b
by FUNCT_1:def 6;
      hence contradiction by A1,A2,A4,A5;
    end;
    hence thesis by A2,A4,FUNCT_1:def 6;
  end;
A6: not x2 c= x1 implies p.:x1 <> p.:x2
  proof
    assume not x2 c= x1;
    then consider a being object such that
A7: a in x2 and
A8: not a in x1;
    not p.a in p.:x1
    proof
      assume p.a in p.:x1;
      then ex b being object st b in dom p & b in x1 & p.a = p.b
by FUNCT_1:def 6;
      hence contradiction by A1,A2,A7,A8;
    end;
    hence thesis by A2,A7,FUNCT_1:def 6;
  end;
  assume x1 <> x2;
  hence thesis by A3,A6,XBOOLE_0:def 10;
end;
