reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;

theorem Th3:
  for i,j be Integer st i < j holds uInt.i < uInt.j
proof
A1:for k be Nat st k>=1 holds uInt.n < uInt.(n+k)
  proof
    defpred P[Nat] means uInt.n < uInt.(n+$1);
A2: P[1] by Lm1;
A3: for j being Nat st 1 <= j holds P[j] implies P[j+1]
    proof
      let j be Nat such that
A4:   1 <= j & P[j];
      uInt.(n+j) <= uInt.(n+j+1) by Lm1;
      hence thesis by A4,SURREALO:4;
    end;
    for j being Nat st 1 <= j holds P[j] from NAT_1:sch 8(A2,A3);
    hence thesis;
  end;
A5:for k be Nat st k>=1 holds uInt.-(n+k) < uInt.-n
  proof
    defpred P[Nat] means uInt.-(n+$1) < uInt.-n;
A6: P[1] by Lm2;
A7: for j being Nat st 1 <= j holds P[j] implies P[j+1]
    proof
      let j be Nat such that
A8:   1 <= j & P[j];
      uInt.-(n+j+1) <= uInt.-(n+j) by Lm2;
      hence thesis by A8,SURREALO:4;
    end;
    for j being Nat st 1 <= j holds P[j] from NAT_1:sch 8(A6,A7);
    hence thesis;
  end;
  let i,j be Integer such that
A9:i < j;
  i in INT by INT_1:def 2;
  then consider I be Nat such that
A10: i=I or i = -I by INT_1:def 1;
  j in INT by INT_1:def 2;
  then consider J be Nat such that
A11:j=J or j = -J by INT_1:def 1;
  per cases by A10,A11;
  suppose
A12:i=I & j=J;
    then reconsider JI=J-I as Nat by A9,NAT_1:21;
    JI <>0 by A12,A9;
    then uInt.I < uInt.(I+JI) by A1,NAT_1:14;
    hence thesis by A12;
  end;
  suppose i=I & j=-J;
    hence thesis by A9;
  end;
  suppose
A13:i=-I & j=J;
    per cases;
    suppose
A14:  i=0;
      then uInt.0 < uInt.(0+j) by A9,A1,NAT_1:14;
      hence thesis by A14;
    end;
    suppose
A15:  j=0;
      then uInt.-(0+I) < uInt.-0 by A9,A13,A5,NAT_1:14;
      hence thesis by A15,A13;
    end;
    suppose i<>0 & j <>0;
      then uInt.-(0+I) < uInt.-0 <= uInt.(0+J) by A13,NAT_1:14,A1,A5;
      hence thesis by A13,SURREALO:4;
    end;
  end;
  suppose
A16: i=-I & j=-J;
    then reconsider IJ=I-J as Nat by A9,XREAL_1:24,NAT_1:21;
    IJ<>0 by A16,A9;
    then uInt.-(J+IJ) < uInt.-J by A5,NAT_1:14;
    hence thesis by A16;
  end;
end;
