 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem
  n <= m implies
    divL(o,Inv).n c= divL(o,Inv).m &
    divR(o,Inv).n c= divR(o,Inv).m
proof
  defpred P[Nat] means
    divL(o,Inv).n c= divL(o,Inv).(n+$1) & divR(o,Inv).n c= divR(o,Inv).(n+$1);
A1: P[0];
A2: for k be Nat st P[k] holds P[k+1]
  proof
    set T=transitions_of(o,Inv);
    let k be Nat such that
A3: P[k];
    set nk=n+k;
A4: divL(o,Inv).(nk+1) = (T.(nk+1))`1 &
    divR(o,Inv).(nk+1) = (T.(nk+1))`2 by Def5,Def6;
A5: divL(o,Inv).nk = L_(T.nk) & divR(o,Inv).nk = R_(T.nk) by Def5,Def6;
    (T.(nk+1))`1 =
    L_(T.nk) \/ divset(L_(T.nk),o,R_o,Inv) \/ divset(R_(T.nk),o,L_o,Inv) &
    (T.(nk+1))`2 =
    R_(T.nk)\/divset(L_(T.nk),o,L_o,Inv)\/divset(R_(T.nk),o,R_o,Inv) by Def4;
    then (T.(nk+1))`1 =
    L_(T.nk) \/ (divset(L_(T.nk),o,R_o,Inv) \/ divset(R_(T.nk),o,L_o,Inv)) &
    (T.(nk+1))`2 =
      R_(T.nk) \/ (divset(L_(T.nk),o,L_o,Inv) \/ divset(R_(T.nk),o,R_o,Inv))
    by XBOOLE_1:4;
    then L_(T.nk) c= L_(T.(nk+1)) & R_(T.nk) c= R_(T.(nk+1)) by XBOOLE_1:7;
    hence thesis by A4,A5,A3,XBOOLE_1:1;
  end;
A6:for k be Nat holds P[k] from NAT_1:sch 2(A1,A2);
  assume n<=m;
  then reconsider mn=m-n as Nat by NAT_1:21;
  m=n+mn;
  hence thesis by A6;
end;
