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

theorem Th31:
  for x be Surreal st 0_No < x
    ex y be Surreal st -1_No == y & sqrt y < -x &
        y = [{},{0_No,(sqrt(x*x+1_No)-x) *(sqrt(x*x+1_No)-x)}]
proof
  let x be Surreal such that
A1: 0_No < x;
  reconsider p=x as positive Surreal by A1,SURREALI:def 8;
A2: -1_No = [-- R_1_No, -- L_1_No] by SURREALR:7
  .= [{}, {-0_No}] by SURREALR:21,22;
A3:0_No <= x by A1;
  then
A4: x*0_No <= x*x by SURREALR:75;
A5: x*x+0_No < x*x+1_No by SURREALR:44,SURREALI:def 8;
  then x == sqrt (x*x) < sqrt (x*x+1_No) by A4, Th27,A3,Th29;
  then x < sqrt (x*x+1_No) by SURREALO:4;
  then 0_No < sqrt (x*x+1_No)-x by SURREALR:45;
  then reconsider Y = sqrt (x*x+1_No)+-x as positive Surreal by SURREALI:def 8;
  set YY=Y*Y;
  {0_No}\/{YY} is surreal-membered;
  then reconsider Y0 = {0_No,YY} as surreal-membered set by ENUMSET1:1;
  consider M be Ordinal such that
A6: o in {}\/Y0 implies ex A be Ordinal st A in M & o in Day A
  by SURREAL0:47;
  {} << Y0;
  then [{},Y0] in Day M by A6,SURREAL0:46;
  then reconsider y = [{},Y0] as Surreal;
  take y;
A7: 0_No <= YY by SURREALI:def 8;
A8:for z be Surreal st z in R_y holds 0_No <= z by A7,TARSKI:def 2;
A9: -1_No = [L_y,{0_No}] by A2;
  0_No in R_y by TARSKI:def 2;
  hence
A10: -1_No == y by A9,A8,SURREALO:24;
  sqrt (YY) +sqrt (YY) is positive;
  then
A11:not sqrt (YY) +sqrt (YY) ==0_No;
  YY in R_y by TARSKI:def 2;
  then YY in R_NonNegativePart y by A7,Def1;
  then sqrt (YY) in R_sqrt_0 y by Def9;
  then sqrt (YY) in sqrtR(sqrt_0 y, y).0 by Th6;
  then
A12: (y + (sqrt (YY))*(sqrt (YY))) * ( sqrt (YY) +sqrt (YY))" in
  sqrt(y,sqrtR(sqrt_0 y, y).0,sqrtR(sqrt_0 y, y).0) by A11,Def2;
  sqrt y = [Union sqrtL(sqrt_0 y,y),Union sqrtR (sqrt_0 y,y)] by Th15;
  then
A13: {sqrt y} << R_sqrt y = Union sqrtR(sqrt_0 y, y) & sqrt y in {sqrt y}
  by TARSKI:def 1,SURREALO:11;
  sqrtR(sqrt_0 y,y).(0+1) = sqrtR(sqrt_0 y,y).0 \/
    sqrt(y,sqrtL(sqrt_0 y,y).0,sqrtL(sqrt_0 y,y).0) \/
    sqrt(y,sqrtR(sqrt_0 y,y).0,sqrtR(sqrt_0 y,y).0) by Th8;
  then
A14: (y + (sqrt (YY))*(sqrt (YY))) * ( sqrt (YY) +sqrt (YY))" in
  sqrtR(sqrt_0 y, y).1 c= Union sqrtR(sqrt_0 y, y)
    by ABCMIZ_1:1,XBOOLE_0:def 3,A12;
  set TWO=1_No + 1_No;
A15: not 0_No == Y" & not 0_No == Y & not TWO == 0_No by SURREALI:def 8;
  then
A16: Y" " == Y & Y * Y"==1_No & TWO*TWO" == 1_No by SURREALI:44,33;
  0_No <= Y by SURREALI:def 8;
  then sqrt YY == Y by Th29;
  then sqrt (YY) +sqrt (YY) == Y + Y = 1_No *Y + 1_No *Y ==
   (1_No + 1_No)*Y by SURREALR:66,67;
  then sqrt (YY) +sqrt (YY) == (1_No + 1_No)*Y by SURREALO:4;
  then ( sqrt (YY) +sqrt (YY))" == ((1_No + 1_No)*Y)" == TWO"*(Y")
  by A11,A15,SURREALI:43,45;
  then
A17:( sqrt (YY) +sqrt (YY))" == TWO"* Y" by SURREALO:4;
A18:(sqrt (YY))*(sqrt (YY)) == YY by A7,Th19;
A19: YY == sqrt (x*x+1_No) * sqrt (x*x+1_No) +
  (-x)*(-x) + (sqrt (x*x+1_No)*(-x) + (-x)*sqrt (x*x+1_No)) by SURREALR:76;
  0_No <= x*x+1_No by A4,A5,SURREALO:4;
  then sqrt (x*x+1_No)*sqrt (x*x+1_No) == x*x+1_No & (-x)*(-x) = x*x
  by Th19,SURREALR:58;
  then
A20: sqrt (x*x+1_No) * sqrt (x*x+1_No) + (-x)*(-x) == x*x + 1_No +x*x
  = 1_No + (x*x+x*x) by SURREALR:37,66;
  x*x = 1_No*(x*x);
  then x*x+x*x == TWO*(x*x) by SURREALR:67;
  then 1_No + (x*x+x*x) == 1_No+TWO*(x*x) by SURREALR:66;
  then
A21: sqrt (x*x+1_No) * sqrt (x*x+1_No) + (-x)*(-x) == 1_No + TWO*(x*x)
  by A20,SURREALO:4;
  1_No*(sqrt (x*x+1_No)*(-x)) = sqrt (x*x+1_No)*(-x);
  then sqrt (x*x+1_No)*(-x) +
  sqrt (x*x+1_No)*(-x) == TWO*(sqrt (x*x+1_No)*(-x)) by SURREALR:67;
  then sqrt (x*x+1_No) * sqrt (x*x+1_No) + (-x)*(-x) +(sqrt (x*x+1_No)*(-x)+
      (-x)*sqrt (x*x+1_No))== 1_No + TWO*(x*x) +TWO*(sqrt (x*x+1_No)*(-x))
  by A21,SURREALR:66;
  then YY == 1_No + TWO*(x*x) +TWO*(sqrt (x*x+1_No)*(-x))
  by A19,SURREALO:4;
  then
A22:(sqrt (YY))*(sqrt (YY)) ==
  1_No + TWO*(x*x) +TWO*(sqrt (x*x+1_No)*(-x))
  = 1_No + (TWO*(x*x) +TWO*(sqrt (x*x+1_No)*(-x)))
  by A18,SURREALO:4,SURREALR:37;
A23: 1_No - 1_No == 0_No by SURREALR:39;
  x*x = (-x)*(-x) by SURREALR:58;
  then TWO*(x*x) +TWO*(sqrt (x*x+1_No)*(-x)) ==
     TWO*(x*x +sqrt (x*x+1_No)*(-x)) == TWO *((-x) * Y) by SURREALR:54,67;
  then TWO*(x*x) +TWO*(sqrt (x*x+1_No)*(-x)) == TWO *((-x) * Y)
  by SURREALO:4;
  then 1_No+ (TWO*(x*x) +TWO*(sqrt (x*x+1_No)*(-x))) == 1_No+
  TWO *((-x) * Y) by SURREALR:66;
  then (sqrt (YY))*(sqrt (YY)) == 1_No +TWO *((-x) * Y)
  by A22,SURREALO:4;
  then y+ (sqrt (YY))*(sqrt (YY)) == -1_No + (1_No +TWO *((-x) * Y))
  = (1_No - 1_No) +TWO *((-x) * Y) == 0_No + TWO *((-x) * Y)
  by A23,A10,SURREALR:66,37;
  then y+ (sqrt (YY))*(sqrt (YY)) == TWO *((-x) * Y)
  by SURREALO:4;
  then (y + (sqrt (YY))*(sqrt (YY))) * ( sqrt (YY) +sqrt (YY))" ==
  (TWO *((-x) * Y)) * (sqrt (YY) +sqrt (YY))"==
  (TWO *((-x) * Y)) * (TWO"* Y") by A17,SURREALR:54;
  then
A24: (y + (sqrt (YY))*(sqrt (YY))) * ( sqrt (YY) +sqrt (YY))" ==
  (TWO *((-x) * Y)) * (TWO"* Y") by SURREALO:4;
  (TWO *((-x) * Y)) * (TWO"* Y") == (-x) * (TWO * Y) * (TWO"* Y")
  == (-x) * ((TWO * Y) * (TWO"* Y")) by SURREALR:69,54;
  then
A25: (TWO *((-x) * Y)) * (TWO"* Y") == (-x) * ((TWO * Y) * (TWO"* Y"))
  by SURREALO:4;
  (TWO * Y) * (TWO"* Y") == (TWO*Y) * TWO"* Y" == Y* (TWO * TWO")* Y"
  by SURREALR:69,54;
  then (TWO * Y) * (TWO"* Y") == Y* (TWO * TWO")* Y" == (TWO * TWO")* (Y*Y")
  by SURREALR:69,SURREALO:4;
  then (TWO * Y) * (TWO"* Y") == (TWO * TWO")* (Y*Y") == (TWO * TWO")* 1_No
  by A16,SURREALR:54,SURREALO:4;
  then (TWO * Y) * (TWO"* Y") == TWO * TWO" by SURREALO:4;
  then (TWO * Y) * (TWO"* Y") == 1_No by A16,SURREALO:4;
  then (-x) * ((TWO * Y) * (TWO"* Y")) == (-x)*1_No by SURREALR:54;
  then (TWO *((-x) * Y)) * (TWO"* Y") == (-x)*1_No by A25,SURREALO:4;
  then (y + (sqrt (YY))*(sqrt (YY))) * (sqrt (YY) +sqrt (YY))" == -x
  by A24,SURREALO:4;
  hence thesis by A14,A13,SURREALO:4;
end;
