reserve x,y,y1,y2,z,e,s for set;
reserve alpha,beta,gamma for Ordinal;
reserve n,m,k for Nat;
reserve g,g0,g1,g2,gO,gL,gR,gLL,gLR,gRL,gRR for ConwayGame;

theorem Th40:
  -(-g) = g
proof
  defpred Bad[ConwayGame] means -(-$1) <> $1;
  assume not thesis;
  then
A1: ex g st Bad[g];
  consider g such that
A2: Bad[g] & for gO st gO in the_Options_of g holds not Bad[gO]
    from ConwayGameMin(A1);

  now
    hereby
      let x be object;
      assume x in the_LeftOptions_of -(-g);
      then consider gR such that
A3:     gR in the_RightOptions_of -g & x = -gR by Th39;
      consider gL such that
A4:     gL in the_LeftOptions_of g & gR = -gL by Th39,A3;
      gL in the_Options_of g by A4,XBOOLE_0:def 3;
      hence x in the_LeftOptions_of g by A2,A3,A4;
    end;
    hereby
      let x be object;
      assume
A5:     x in the_LeftOptions_of g;
      reconsider gL = x as ConwayGame by Th18,A5;
      -gL in the_RightOptions_of -g & gL in the_Options_of g
        by Th39,A5,XBOOLE_0:def 3;
      then -(-gL) in the_LeftOptions_of -(-g) & -(-gL) = gL
        by Th39,A2;
      hence x in the_LeftOptions_of -(-g);
    end;
  end;
  then
A6: the_LeftOptions_of -(-g) = the_LeftOptions_of g by TARSKI:2;

  now
    hereby
      let x be object;
      assume x in the_RightOptions_of -(-g);
      then consider gL such that
A7:     gL in the_LeftOptions_of -g & x = -gL by Th39;
      consider gR such that
A8:     gR in the_RightOptions_of g & gL = -gR by Th39,A7;
      gR in the_Options_of g by A8,XBOOLE_0:def 3;
      hence x in the_RightOptions_of g by A2,A7,A8;
    end;
    hereby
      let x be object;
      assume
A9:     x in the_RightOptions_of g;
      reconsider gR = x as ConwayGame by Th18,A9;
      -gR in the_LeftOptions_of -g & gR in the_Options_of g
        by Th39,A9,XBOOLE_0:def 3;
      then -(-gR) in the_RightOptions_of -(-g) & -(-gR) = gR
        by Th39,A2;
      hence x in the_RightOptions_of -(-g);
    end;
  end;
  then the_RightOptions_of -(-g) = the_RightOptions_of g by TARSKI:2;
  hence contradiction by A2,A6,Th5;
end;
