reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;

theorem
  z = [{x,y},X] implies [{x},X] is Surreal
proof
  set b = born z;
  assume A1:z = [{x,y},X];
  A2: z in Day b by SURREAL0:def 18;
  then A3:{x,y} << X & for o be object st o in {x,y} \/ X
  ex O st O in b & o in Day O by A1,SURREAL0:46;
  A4:{x} << X
  proof
    let l,r;
    assume A5:l in {x} & r in X;
    then l = x by TARSKI:def 1;
    then l in {x,y} by TARSKI:def 2;
    hence thesis by A3,A5;
  end;
  for o be object st o in {x} \/ X ex O st O in b & o in Day O
  proof
    let o be object;
    assume o in {x} \/ X;
    then o in {x} or o in X by XBOOLE_0:def 3;
    then o =x or o in X by TARSKI:def 1;
    then o in {x,y} or o in X by TARSKI:def 2;
    then o in {x,y} \/ X by XBOOLE_0:def 3;
    hence thesis by A2,A1,SURREAL0:46;
  end;
  then [{x},X] in Day b by A4,SURREAL0:46;
  hence thesis;
end;
