 reserve n,m for Nat,
      o for object,
      p for pair object,
      x,y,z for Surreal;

theorem Th7:
  n <= m implies
     sqrtL(p,o).n c= sqrtL(p,o).m &
     sqrtR(p,o).n c= sqrtR(p,o).m
proof
  defpred P[Nat] means sqrtL(p,o).n c= sqrtL(p,o).(n+$1) &
     sqrtR(p,o).n c= sqrtR(p,o).(n+$1);
A1:P[0];
A2:for k be Nat st P[k] holds P[k+1]
  proof
    set T=transitions_of(p,o);
    let k be Nat such that
A3: P[k];
    set nk=n+k;
A4: sqrtL(p,o).(nk+1) = (T.(nk+1))`1 &
    sqrtR(p,o).(nk+1) = (T.(nk+1))`2 by Def4,Def5;
A5: sqrtL (p,o).nk = L_(T.nk) & sqrtR(p,o).nk = R_(T.nk) by Def4,Def5;
    (T.(nk+1))`1 = L_(T.nk) \/ sqrt(o,L_(T.nk),R_(T.nk)) &
    (T.(nk+1))`2 = R_(T.nk) \/ sqrt(o,L_(T.nk),L_(T.nk)) \/
    sqrt(o,R_(T.nk),R_(T.nk)) by Def3;
    then (T.(nk+1))`1 = L_(T.nk) \/ sqrt(o,L_(T.nk),R_(T.nk)) &
    (T.(nk+1))`2 = R_(T.nk) \/ (sqrt(o,L_(T.nk),L_(T.nk)) \/
    sqrt(o,R_(T.nk),R_(T.nk))) 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;
