reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;

theorem Th3:
  x <= x
proof
   defpred P[Ordinal] means for x be Surreal st x in Day $1 holds x <= x;
   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];
     let x be Surreal such that A3: x in Day D;
     A4:x= [L_x,R_x];
     assume not x<=x;
     then per cases by SURREAL0:43;
     suppose not L_x << {x};
       then consider xl,r be Surreal such that
       A5: xl in L_x & r in {x} & r <= xl;
       xl in L_x\/R_x by A5,XBOOLE_0:def 3;
       then consider O be Ordinal such that
       A6: O in D & xl in Day O by A4,A3,SURREAL0:46;
       x <= xl by A5,TARSKI:def 1;
       then A7: L_x << {xl} by SURREAL0:43;
       xl in {xl} by TARSKI:def 1;
       hence thesis by A6,A2,A5,A7;
     end;
     suppose not {x} << R_x;
       then consider l,xr be Surreal such that
       A8: l in {x} & xr in R_x & xr <= l;
       xr in L_x\/R_x by A8,XBOOLE_0:def 3;
       then consider O be Ordinal such that
       A9: O in D & xr in Day O by A4,A3,SURREAL0:46;
       xr <= x by A8,TARSKI:def 1;
       then A10: {xr} << R_x by SURREAL0:43;
       xr in {xr} by TARSKI:def 1;
       hence thesis by A9,A2,A8,A10;
     end;
   end;
   A11:for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
   ex A be Ordinal st x in Day A by SURREAL0:def 14;
   hence thesis by A11;
end;
