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

theorem Th18:
  x <= 0_No implies Union sqrtL(sqrt_0 x,o) = {}
proof
  assume
A1: x<=0_No;
  defpred P[Nat] means sqrtL(sqrt_0 x,o).$1={};
A2:P[0]
  proof
    assume sqrtL(sqrt_0 x,o).0<>{};
    then consider a be object such that
A3: a in sqrtL(sqrt_0 x,o).0 by XBOOLE_0:def 1;
    a in L_sqrt_0 x by A3,Th6;
    then consider l be Surreal such that
A4: a = sqrt l & l in L_NonNegativePart x by Def9;
A5: l in L_x & 0_No <= l by A4,Th2;
    L_x << {x} & x in {x} by TARSKI:def 1,SURREALO:11;
    hence thesis by A5,A1,SURREALO:4;
  end;
A6:P[n] implies P[n+1]
  proof
    assume
A7: P[n];
    assume sqrtL(sqrt_0 x,o).(n+1)<>{};
    then consider y be object such that
A8: y in sqrtL(sqrt_0 x,o).(n+1) by XBOOLE_0:def 1;
    y in sqrtL(sqrt_0 x,o).n \/ sqrt(o,sqrtL(sqrt_0 x,o).n,sqrtR(sqrt_0 x,o).n)
    by A8,Th8;
    then per cases by XBOOLE_0:def 3;
    suppose y in sqrtL(sqrt_0 x,o).n;
      hence thesis by A7;
    end;
    suppose y in sqrt(o,sqrtL(sqrt_0 x,o).n,sqrtR(sqrt_0 x,o).n);
      then ex x1,y1 be Surreal st
      x1 in sqrtL(sqrt_0 x,o).n & y1 in sqrtR(sqrt_0 x,o).n &
      not x1 + y1 == 0_No & y = (o +'(x1 * y1)) * (x1+y1)" by Def2;
      hence thesis by A7;
    end;
  end;
A9:P[n] from NAT_1:sch 2(A2,A6);
  assume Union sqrtL(sqrt_0 x,o) <>{};
  then consider a be object such that
A10:a in Union sqrtL(sqrt_0 x,o) by XBOOLE_0:def 1;
  consider n be object such that
A11:n in dom sqrtL(sqrt_0 x,o) & a in sqrtL(sqrt_0 x,o).n by A10,CARD_5:2;
  dom sqrtL(sqrt_0 x,o)=NAT by Def4;
  then reconsider n as Nat by A11;
  sqrtL(sqrt_0 x,o).n ={} by A9;
  hence thesis by A11;
end;
