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

theorem Th4:
  x <= y & y <= z implies x <= z
proof
  defpred P[Ordinal] means
     for x,y,z be Surreal st x <= y & y <= z &
     born x (+) born y (+) born z c= $1 holds x <= z;
  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,y,z be Surreal such that
    A3:x <= y & y <= z and
    A4:born x (+) born y (+) born z c= D;
    per cases by A4,ORDINAL1:11,XBOOLE_0:def 8;
    suppose born x (+) born y (+) born z in D;
      hence thesis by A2,A3;
    end;
    suppose A5:born x (+) born y (+) born z = D;
      assume not x <= z;
      then per cases by SURREAL0:43;
      suppose not L_x << {z};
        then consider xl,r be Surreal such that
        A6: xl in L_x & r in {z} & r <= xl;
        A7: z <= xl by A6,TARSKI:def 1;
        xl in L_x \/ R_x by A6,XBOOLE_0:def 3;
        then born xl (+) born y in born x (+) born y by Th1,ORDINAL7:94;
        then born xl (+) born y  (+) born z in D by A5,ORDINAL7:94;
        then A8: born y  (+) born z (+) born xl in D by ORDINAL7:68;
        L_x << {y} & y in {y} by A3,SURREAL0:43,TARSKI:def 1;
        hence thesis by A6,A8, A7,A3,A2;
      end;
      suppose not {x} << R_z;
        then consider l,zr be Surreal such that
        A9: l in {x}& zr in R_z & zr <= l;
        A10:  zr <= x by A9,TARSKI:def 1;
        zr in L_z \/ R_z by A9,XBOOLE_0:def 3;
        then born zr (+) born x in born z (+) born x by Th1,ORDINAL7:94;
        then born zr (+) born x (+) born y in born z (+) born x (+) born y
        by ORDINAL7:94;
        then A11: born zr (+) born x (+) born y in D by A5,ORDINAL7:68;
        {y} << R_z & y in {y} by A3,SURREAL0:43,TARSKI:def 1;
        hence thesis by A9,A11,A2,A3,A10;
      end;
    end;
  end;
  A12:for D be Ordinal holds P[D] from ORDINAL1:sch 2(A1);
  P[ born x (+) born y (+) born z] by A12;
  hence thesis;
end;
