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

theorem Th23:
  for Max be Surreal st
    (for y st y in L_x holds y <= Max) & Max in L_x holds
       [{Max},R_x] is Surreal &
       for y st y = [{Max},R_x] holds y == x & born y c= born x
proof
  let Max be Surreal such that
  A1:(for y st y in L_x holds y <= Max) and A2: Max in L_x;
  A3: L_x << R_x by SURREAL0:45;
  A4: {Max} << R_x
  proof
    let l,r such that A5:l in {Max} & r in R_x;
    l = Max by A5,TARSKI:def 1;
    hence thesis by A3,A5,A2;
  end;
  for o be object st o in {Max} \/ R_x ex O st O in born x & o in Day O
  proof
    let o be object such that A6:o in {Max} \/ R_x;
    o = Max or o in R_x by A6,ZFMISC_1:136;
    then A7:o in L_x \/ R_x by A2,XBOOLE_0:def 3;
    reconsider o as Surreal by SURREAL0:def 16,A6;
    take born o;
    thus thesis by SURREAL0:def 18,A7,Th1;
  end;
  then A8: [{Max},R_x] in Day born x by A4,SURREAL0:46;
  hence [{Max},R_x] is Surreal;
  let y such that A9: y = [{Max},R_x];
  A10:x=[L_x,R_x] & y = [L_y,R_y];
  A11:for x1 be Surreal st x1 in L_x ex y1 be Surreal st y1 in L_y & x1 <= y1
  proof
    let x1 be Surreal such that A12: x1 in L_x;
    take Max;
    thus thesis by A12,A1,A9,TARSKI:def 1;
  end;
  A13: for x1 be Surreal st x1 in R_y
  ex y1 be Surreal st y1 in R_x & y1 <= x1 by A9;
  A14:for x1 be Surreal st x1 in L_y ex y1 be Surreal st y1 in L_x & x1 <= y1
  proof
    let x1 be Surreal such that A15: x1 in L_y;
    take Max;
    thus thesis by A15,A2,A9,TARSKI:def 1;
  end;
  for x1 be Surreal st x1 in R_x ex y1 be Surreal st y1 in R_y & y1 <= x1
  by A9;
  hence thesis by SURREAL0:def 18,A9,A8,A13, A10,A14,SURREAL0:44,A11;
end;
