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 Th69:
  x in Day A implies x <= No_Ordinal_op A
proof
  defpred P[Ordinal] means for x st x in Day $1 holds
  x <= No_Ordinal_op $1;
A1: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
  proof
    let D be Ordinal such that
A2: for C be Ordinal st C in D holds P[C];
    set O = No_Ordinal_op D;
    let x;
    assume
A3: x in Day D;
    L_x << {O}
    proof
      let a,b be Surreal such that
A4:   a in L_x & b in {O};
A5:   a in L_x\/R_x by XBOOLE_0:def 3,A4;
      then
A6:   born a in born x by SURREALO:1;
A7:   No_Ordinal_op born a < No_Ordinal_op born x by A5,SURREALO:1,Th68;
A8:   born x c= D by A3,SURREAL0:def 18;
      then
A9:   No_Ordinal_op born x <= O by Th68,ORDINAL1:5;
      a in Day born a by SURREAL0:def 18;
      then a <= No_Ordinal_op born a by A2,A6,A8;
      then a < No_Ordinal_op born x by A7,SURREALO:4;
      then a < O by A9,SURREALO:4;
      hence thesis by A4,TARSKI:def 1;
    end;
    then L_x << {O} & {x}<<R_O by Def10;
    hence thesis by SURREAL0:43;
  end;
  for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
  hence thesis;
end;
