reserve A,A1,A2,B,B1,B2,C,O for Ordinal,
      R,S for Relation,
      a,b,c,o,l,r for object;
reserve x,y,z,t,r,l for Surreal,
        X,Y,Z for set;

theorem Th45:
  L_x << R_x
proof
  let l,r be Surreal such that
  A1: l in L_x & r in R_x;
  consider A be Ordinal such that
  A2: x in Day A by Def14;
  set S=No_Ord A;
  A3: L_x <<S,R_x by A2,Th7;
  l in L_x \/R_x by A1,XBOOLE_0:def 3;
  then consider OL be Ordinal such that
  A4:OL in A & l in Day(S,OL) by A2,Th7;
  OL c= A by A4,ORDINAL1:def 2;
  then A5:l in Day OL c= Day A by A4,Th35,Th36;
  r in L_x \/R_x by A1,XBOOLE_0:def 3;
  then consider OR be Ordinal such that
  A6:OR in A & r in Day(S,OR) by A2,Th7;
  OR c= A by A6,ORDINAL1:def 2;
  then r in Day OR c= Day A by A6,Th35,Th36;
  hence thesis by A3,A1,A5,Th40;
end;
