reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;
reserve r,r1,r2 for Real;

theorem
  ex x be No_ordinal Surreal st
      born x = A & No_uOrdinal_op A == x &
     (for o holds o in L_x iff ex B st B in A & o = No_uOrdinal_op B)
proof
  defpred P[object] means ex B st B in A & $1 = No_uOrdinal_op B;
  consider X be set such that
A1: o in X iff o in Day A & P[o] from XBOOLE_0:sch 1;
A2:X<<{};
  o in X \/ {} implies ex O st O in A & o in Day O
  proof
    assume o in X \/ {};
    then consider B such that
A3: B in A & o = No_uOrdinal_op B by A1;
    born No_uOrdinal_op B = B by Th73;
    then No_uOrdinal_op B in Day B by SURREAL0:def 18;
    hence thesis by A3;
  end;
  then
A4: [X,{}] in Day A by A2,SURREAL0:46;
  then reconsider x= [X,{}] as Surreal;
  R_x = {};
  then reconsider x as No_ordinal Surreal by Def10;
  take x;
  for O st x in Day O holds A c= O
  proof
    let O such that
A5: x in Day O & not A c= O;
A6: No_uOrdinal_op O in Day O by Th74;
    Day O c= Day A by A5,SURREAL0:35;
    then No_uOrdinal_op O in X=L_x<<{x} & x in {x}
    by A6,A5,ORDINAL1:16,A1,SURREALO:11,TARSKI:def 1;
    hence thesis by A5,Th76;
  end;
  hence born x = A by A4,SURREAL0:def 18;
  L_(No_Ordinal_op A) << {x}
  proof
    let a,b be Surreal such that
A7: a in L_(No_Ordinal_op A) & b in {x};
    consider B such that
A8: B in A & a = No_Ordinal_op B by A7,Th71;
A9: No_uOrdinal_op B in Day B c= Day A
    by A8,ORDINAL1:def 2,Th74,SURREAL0:35;
    No_uOrdinal_op B in X =L_x<<{x} by A8,A1,SURREALO:11,A9;
    then a == No_uOrdinal_op B < b by A8,Th73,A7;
    hence thesis by SURREALO:4;
  end;
  then
A10: L_(No_Ordinal_op A) << {x} & {No_Ordinal_op A}<< R_x;
  L_x << {No_Ordinal_op A}
  proof
    let a,b be Surreal such that
A11:a in L_x & b in {No_Ordinal_op A};
    consider B such that
A12:B in A & a = No_uOrdinal_op B by A1,A11;
    a < No_uOrdinal_op A == No_Ordinal_op A by A12,Th75,Th73;
    then a < No_Ordinal_op A by SURREALO:4;
    hence thesis by A11,TARSKI:def 1;
  end;
  then L_x << {No_Ordinal_op A} & {x} << R_(No_Ordinal_op A) by Def10;
  then No_uOrdinal_op A == No_Ordinal_op A == x by SURREAL0:43,A10,Th73;
  hence No_uOrdinal_op A==x by SURREALO:4;
  let o;
  thus o in L_x implies ex B st B in A & o = No_uOrdinal_op B by A1;
  given B such that
A13: B in A & o = No_uOrdinal_op B;
  o in Day B c= Day A by A13,ORDINAL1:def 2,Th74,SURREAL0:35;
  hence thesis by A13,A1;
end;
