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

theorem
  z = [X,{x,y}] implies [X,{x}] is Surreal
proof
  set b = born z;
  assume A1:z = [X,{x,y}];
  A2:z in Day b by SURREAL0:def 18;
  then A3:X<<{x,y} & for o be object st o in X\/{x,y}
    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 r = x by TARSKI:def 1;
    then r 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;
