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 Th40:
  x <= y iff
    for A st x in Day A & y in Day A holds x <=No_Ord A,y
proof
  consider Ax be Ordinal such that
  A1:x in Day Ax by Def14;
  consider Ay be Ordinal such that
  A2:y in Day Ay by Def14;
  thus x <= y implies
  for A be Ordinal st x in Day A & y in Day A
  holds x <=No_Ord A,y by Th39;
  assume A3:for A be Ordinal st x in Day A & y in Day A
    holds x <=No_Ord A,y;
  set A=Ax\/Ay;
  Day Ax c= Day A & Day Ay c= Day A by Th35,XBOOLE_1:7;
  hence thesis by A1,A2,A3;
end;
