reserve x,x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,y for object, X,Z for set;

theorem
  { x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 } = { x1,x2,x3,x4,x5,x6,x7,x8,x9 } \/
  { x10 }
proof
  now
    let x be object;
A1: x in {x10} iff x=x10 by TARSKI:def 1;
    x=x1 or x=x2 or x=x3 or x=x4 or x=x5 or x=x6 or x=x7 or x=x8 or x=x9
    or x=x10 iff x in { x1,x2,x3,x4,x5,x6,x7,x8,x9 } or x = x10 by Def7;
    hence
    x in { x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 } iff x in { x1,x2,x3,x4,x5,x6,
    x7,x8,x9 } \/ {x10} by A1,Def8,XBOOLE_0:def 3;
  end;
  hence thesis by TARSKI:2;
end;
