reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;

theorem
  for X being set st card X = 2
     ex x,y being object st x <> y & X = {x,y}
proof
  let X be set;
  assume
A1: card X = 2;
  then consider x being object such that
A2: x in X by CARD_1:27,XBOOLE_0:def 1;
  X is finite by A1;
  then reconsider Y = X as finite set;
  {x} c= X by A2,ZFMISC_1:31;
  then card(X \ {x}) = card Y - card{x} by Th43
    .= 2 - 1 by A1,CARD_1:30;
  then consider y being object such that
A3: X \ {x} = {y} by Th41;
  take x,y;
  x in {x} by TARSKI:def 1;
  hence x <> y by A3,XBOOLE_0:def 5;
  thus X c= {x,y}
  proof
    let z be object;
    assume
A4: z in X;
    per cases;
    suppose
      z = x;
      hence thesis by TARSKI:def 2;
    end;
    suppose
      z <> x;
      then not z in {x} by TARSKI:def 1;
      then z in {y} by A3,A4,XBOOLE_0:def 5;
      then z = y by TARSKI:def 1;
      hence thesis by TARSKI:def 2;
    end;
  end;
  let z be object;
  assume z in {x,y};
  then
A5: z = x or z = y by TARSKI:def 2;
  y in X \ {x} by A3,TARSKI:def 1;
  hence thesis by A2,A5;
end;
