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

theorem Th20:
  x <= 0_No implies sqrt x is surreal
proof
  assume
A1: x <= 0_No;
A2: L_sqrt_0 x is surreal-membered
  proof
    let o;
    assume o in L_sqrt_0 x;
    then consider l be Surreal such that
A3: o = sqrt l & l in L_NonNegativePart x by Def9;
    0_No <= l by A3,Th2;
    hence thesis by A3,Th19;
  end;
  R_sqrt_0 x is surreal-membered
  proof
    let o;
    assume o in R_sqrt_0 x;
    then consider l be Surreal such that
A4: o = sqrt l & l in R_NonNegativePart x by Def9;
    0_No <= l by A4,Th2;
    hence thesis by A4,Th19;
  end;
  then Union sqrtR(sqrt_0 x,x) is surreal-membered &
    Union sqrtL(sqrt_0 x,x) is surreal-membered by A2,Th10;
  then consider M be Ordinal such that
A5:for o st o in Union sqrtL(sqrt_0 x,x)\/Union sqrtR(sqrt_0 x,x)
          ex A be Ordinal st A in M & o in Day A by SURREAL0:47;
  Union sqrtL(sqrt_0 x,x) <<Union sqrtR(sqrt_0 x,x) by A1,Th18;
  then [Union sqrtL(sqrt_0 x,x),Union sqrtR(sqrt_0 x,x)] in Day M
  by A5,SURREAL0:46;
  hence sqrt x is surreal by Th15;
end;
