 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem Th34:
  for X,nX be set, Y be surreal-membered set
     st x is positive &
    ((X = L_x & nX=L_||.x.||) or (X=R_x & nX=R_||.x.||))
  holds
     divset(X,||.x.||,Y) = divset(Y,||.x.||,nX,No_inverses_on ||.x.||)
proof
  set Nx = ||.x.||,Inv = No_inverses_on Nx;
  let X,X1 be set, Y be surreal-membered set such that
A1: x is positive and
A2: ((X = L_x & X1=L_||.x.||) or (X=R_x & X1=R_||.x.||));
  thus divset(X,Nx,Y) c= divset(Y,Nx,X1,Inv)
  proof
    let o;
    assume o in divset(X,Nx,Y);
    then consider x1,y1 be Surreal such that
A3: 0_No < x1 & x1 in X & y1 in Y and
A4: o = (1_No +(x1-Nx)*y1)* (x1") by Def15;
A5: x1 is positive by A3;
    not x1==0_No by A3;
    then
A6: x1"=inv x1 by A5,Def14;
A7: x1 is positive by A3;
A8: 0_No <=0_No;
A9: x1 in X1 by A2,A3,A1,A7,Def9;
    x1 in L_Nx or x1 in R_Nx by A2,A3,A1,A7,Def9;
    then x1 in (L_Nx\/R_Nx) by XBOOLE_0:def 3;
    then x1 in (L_Nx\/R_Nx)\{0_No} by A8,A3,ZFMISC_1:56;
    then o = (1_No +'(x1+'-'Nx)*'y1)*' (Inv.x1) by A6,A4,Def13;
    then o in divs(y1,Nx,X1,Inv) by A9,A8,A3,Def2;
    hence thesis by A3,Def3;
  end;
  let o;
  assume o in divset(Y,Nx,X1,Inv);
  then consider y1 be object such that
A10:y1 in Y & o in divs(y1,Nx,X1,Inv) by Def3;
  reconsider y1 as Surreal by A10,SURREAL0:def 16;
  consider x1 be object such that
A11:x1 in X1 & x1 <> 0_No &
    o = (1_No +'(x1 +' -' Nx) *' y1) *' (Inv.x1)
    by A10,Def2;
  reconsider x1 as  Surreal by A2,A11,SURREAL0:def 16;
A12:x1 in X & x1 is positive by A2,A1,A11,Def9;
A13:not x1==0_No by A12;
A14: x1" = inv x1 by A12,A13,Def14;
  x1 in (L_Nx\/R_Nx) by XBOOLE_0:def 3,A2,A11;
  then x1 in (L_Nx\/R_Nx)\{0_No} by A11,ZFMISC_1:56;
  then o = (1_No +(x1 - Nx) * y1) * (x1") by A11,A14,Def13;
  hence thesis by Def15,A12,A10;
end;
