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

theorem Th24:
  for Min be Surreal st
   (for y st y in R_x holds  Min <= y) & Min in R_x holds
       [L_x,{Min}] is Surreal &
       for y st y = [L_x,{Min}] holds y == x & born y c= born x
proof
  let Min be Surreal such that
  A1:(for y st y in R_x holds Min <= y) and A2: Min in R_x;
  A3: L_x << R_x by SURREAL0:45;
  A4: L_x << {Min}
  proof
    let l,r such that A5:l in L_x & r in {Min} ;
    r = Min by A5,TARSKI:def 1;
    hence thesis by A3,A5,A2;
  end;
  for o be object st o in L_x \/ {Min} ex O st O in born x & o in Day O
  proof
    let o be object such that A6:o in L_x \/ {Min};
     o = Min or o in L_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 A7,Th1,SURREAL0:def 18;
  end;
  then A8: [L_x,{Min}] in Day born x by A4,SURREAL0:46;
  hence [L_x,{Min}] is Surreal;
  let y such that A9: y = [L_x,{Min}];
  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
  by A9;
  A12: for x1 be Surreal st x1 in R_y ex y1 be Surreal st y1 in R_x & y1 <= x1
  proof
    let x1 be Surreal such that A13: x1 in R_y;
    take Min;
    thus thesis by A9,A13,A2,TARSKI:def 1;
  end;
  A14:for x1 be Surreal st x1 in L_y ex y1 be Surreal st y1 in L_x & x1 <= y1
  by A9;
  for x1 be Surreal st x1 in R_x ex y1 be Surreal st y1 in R_y & y1 <= x1
  proof
    let x1 be Surreal such that A15: x1 in R_x;
    take Min;
    thus thesis by A15,A1,A9,TARSKI:def 1;
  end;
  hence thesis by A10,A14,SURREAL0:44,A12,A11,SURREAL0:def 18,A9,A8;
end;
