
theorem Th8:
  for X being non empty set, x,y being Element of X for f being
  Enumeration of X holds f+*(x,f.y)+*(y,f.x) is Enumeration of X
proof
  let X be non empty set, x,y be Element of X;
  let f be Enumeration of X;
  set g = f+*(x,f.y)+*(y,f.x);
  set A = dom g;
A1: dom (f+*(x,f.y)) = dom f by FUNCT_7:30;
A2: A = dom (f+*(x,f.y)) by FUNCT_7:30;
A3: dom f = X by Th6;
A4: rng f = card X by Th6;
A5: rng (f+*(x,f.y)+*(y,f.x)) = rng f
  proof
    {f.x} c= rng f by A4,ZFMISC_1:31;
    then
A6: rng f \/ {f.x} = rng f by XBOOLE_1:12;
A7: rng g c= rng (f+*(x,f.y)) \/ {f.x} by FUNCT_7:100;
    {f.y} c= rng f by A4,ZFMISC_1:31;
    then rng f \/ {f.y} = rng f by XBOOLE_1:12;
    then rng (f+*(x,f.y)) \/ {f.x} c= rng f by A6,FUNCT_7:100,XBOOLE_1:9;
    hence rng g c= rng f by A7;
    let z be object;
    assume z in rng f;
    then consider a being object such that
A8: a in dom f and
A9: z = f.a by FUNCT_1:def 3;
    per cases;
    suppose
A10:  a <> x & a <> y;
      then
A11:  g.a = (f+*(x,f.y)).a by FUNCT_7:32;
      (f+*(x,f.y)).a = z by A9,A10,FUNCT_7:32;
      hence thesis by A1,A2,A8,A11,FUNCT_1:def 3;
    end;
    suppose
      a = x;
      then g.y = z by A1,A3,A9,FUNCT_7:31;
      hence thesis by A1,A2,A3,FUNCT_1:def 3;
    end;
    suppose
A12:  a = y;
      then
A13:  x <> y implies g.x = (f+*(x,z)).x by A9,FUNCT_7:32;
A14:  (f+*(x,z)).x = z by A3,FUNCT_7:31;
      x = y implies g.x = z by A1,A3,A9,A12,FUNCT_7:31;
      hence thesis by A1,A2,A3,A14,A13,FUNCT_1:def 3;
    end;
  end;
  f+*(x,f.y)+*(y,f.x) is one-to-one
  proof
    let a,b being object;
A15: a <> y implies g.a = (f+*(x,f.y)).a by FUNCT_7:32;
A16: a <> x implies (f+*(x,f.y)).a = f.a by FUNCT_7:32;
A17: b = y implies g.b = f.x by A1,A3,FUNCT_7:31;
A18: b <> y implies g.b = (f+*(x,f.y)).b by FUNCT_7:32;
A19: b = x implies (f+*(x,f.y)).b = f.y by A3,FUNCT_7:31;
A20: a = x implies (f+*(x,f.y)).a = f.y by A3,FUNCT_7:31;
A21: b <> x implies (f+*(x,f.y)).b = f.b by FUNCT_7:32;
    a = y implies g.a = f.x by A1,A3,FUNCT_7:31;
    hence thesis by A2,A3,A15,A20,A16,A17,A18,A19,A21,FUNCT_1:def 4;
  end;
  hence thesis by A1,A2,A3,A4,A5,Th6;
end;
