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 Th39:
  a <=No_Ord A,b & a in Day B & b in Day B
    implies a <=No_Ord B,b
proof
  set R=No_Ord A, S=No_Ord B;
  assume A1: a <=R,b & a in Day B & b in Day B;
  then A2:[a,b] in ClosedProd(S,B,B) by Th33;
  per cases;
  suppose A3:B c= A;
    then ClosedProd(No_Ord B,B,B) = ClosedProd(No_Ord A,B,B) by Th32;
    then No_Ord B = No_Ord A /\ ClosedProd(No_Ord B,B,B) by A3,Th34;
    hence thesis by A1,A2,XBOOLE_0:def 4;
  end;
  suppose A c= B;
    then A4:ClosedProd(S,A,A) c= ClosedProd(S,B,B) by Th30;
    A5: [:Day(S,B),Day(S,B):] = ClosedProd(S,B,B)&
  [:Day(R,A),Day(R,A):] = ClosedProd(R,A,A) by Lm3;
    then R preserves_No_Comparison_on ClosedProd(R,A,A) &
    S preserves_No_Comparison_on ClosedProd(S,B,B) by Def12;
    then R preserves_No_Comparison_on ClosedProd(R,A,A) &
    S preserves_No_Comparison_on ClosedProd(S,A,A) by A4;
    then A6:R /\ ClosedProd(R,A,A) = S /\ ClosedProd(S,A,A) by Th23;
    A7:No_Ord A c= ClosedProd(R,A,A) by A5,Def12;
    [a,b] in R/\ClosedProd(R,A,A) by A1,A7,XBOOLE_0:def 4;
    hence thesis by A6,XBOOLE_0:def 4;
  end;
end;
