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 Th43:
  x <= y iff L_x << {y} & {x} << R_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;
  set A=Ax\/Ay;
  A3: Day Ax c= Day A & Day Ay c= Day A by XBOOLE_1:7,Th35;
  then A4:x in Day A & y in Day A by A1,A2;
  set S = No_Ord A;
  [:Day(S,A),Day(S,A):] = ClosedProd(S,A,A) by Lm3; then
  A5: S preserves_No_Comparison_on ClosedProd(S,A,A) &
  S c= ClosedProd(S,A,A) by Def12;
  thus x <= y implies L_x << {y} & {x} << R_y
  proof
    assume x <= y;
    then x<= S,y by A3,A1,A2,Th40;
    then A6:L_x <<S, {y} & {x} <<S, R_y by A5;
    thus L_x << {y}
    proof
      given l,r be Surreal such that
      A7:  l in L_x & r in {y} & r <= l;
      A8: r=y by A7,TARSKI:def 1;
      l in L_x \/ R_x by A7,XBOOLE_0:def 3;
      then consider Ol be Ordinal such that
      A9: Ol in A & l in Day(S,Ol) by A4,Th7;
      Day(S,Ol) c= Day(S,A) by Th9,A9,ORDINAL1:def 2;
      hence thesis by A9,A6,A3,A2,A8,A7,Th40;
    end;
    thus {x} << R_y
    proof
      given l,r be Surreal such that
      A10: l in {x} & r in R_y & r <= l;
      A11: l=x by A10,TARSKI:def 1;
      r in L_y \/ R_y by A10,XBOOLE_0:def 3;
      then consider Or be Ordinal such that
      A12: Or in A & r in Day(S,Or) by A4,Th7;
      Day(S,Or) c= Day(S,A) by Th9,A12,ORDINAL1:def 2;
      hence thesis by A3,A12,A6,A1,A11,A10,Th40;
    end;
  end;
  assume A13:L_x << {y} & {x} << R_y;
  A14: [x,y] in ClosedProd(S,A,A) by A3,A1,A2,Th33;
  A15: L_x <<S, {y}
  proof
    given l,r be object such that
    A16: l in L_x & r in {y} & l >=S, r;
    [r,l] in ClosedProd(S,A,A) by A16,A5;
    then l in Day A & r in Day A by ZFMISC_1:87;
    then reconsider l,r as Surreal;
    r <= l by A16;
    hence thesis by A13,A16;
  end;
  {x} <<S, R_y
  proof
    given l,r be object such that
    A17: l in {x} & r in R_y & l >=S, r;
    [r,l] in ClosedProd(S,A,A) by A17,A5;
    then l in Day A & r in Day A by ZFMISC_1:87;
    then reconsider l,r as Surreal;
    r <= l by A17;
    hence thesis by A13,A17;
  end;
  then x<= S,y by A15,A14,A5;
  hence x <= y;
end;
