 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 Th16:
  for x be object st (L_x \/ R_x)\{0_No} c= Z & I1|Z = I2|Z holds
    transitions_of(x,I1) = transitions_of(x,I2)
proof
  let x be object such that
A1: (L_x \/ R_x)\{0_No} c= Z & I1|Z = I2|Z;
  set T1 = transitions_of(x,I1),T2 = transitions_of(x,I2);
  defpred P[Nat] means
  T1.$1 = T2.$1;
  T1.0 = 1_No by Def4;
  then
A2:  P[0] by Def4;
A3: L_x\{0_No} c= (L_x \/ R_x)\{0_No} &
  R_x\{0_No} c= (L_x \/ R_x)\{0_No} by XBOOLE_1:7,33;
A4:P[n] implies P[n+1]
  proof
    assume
A5: P[n];
A6: T1.(n+1) is pair & T2.(n+1) is pair by Def4;
A7: (T1.(n+1))`1 = L_(T1.n) \/ divset(L_(T1.n),x,R_x,I1) \/
    divset(R_(T1.n),x,L_x,I1) by Def4
    .= L_(T1.n) \/ divset(L_(T1.n),x,R_x,I2) \/ divset(R_(T1.n),x,L_x,I1)
    by A1,A3,XBOOLE_1:1,Th15
    .= L_(T2.n) \/ divset(L_(T2.n),x,R_x,I2) \/ divset(R_(T2.n),x,L_x,I2)
    by A1,A3,XBOOLE_1:1,Th15,A5
    .= (T2.(n+1))`1 by Def4;
    (T1.(n+1))`2 =R_(T1.n) \/ divset(L_(T1.n),x,L_x,I1) \/
    divset(R_(T1.n),x,R_x,I1) by Def4
    .= R_(T1.n) \/ divset(L_(T1.n),x,L_x,I1) \/ divset(R_(T1.n),x,R_x,I2)
    by A1,A3,XBOOLE_1:1,Th15
    .= R_(T2.n) \/ divset(L_(T2.n),x,L_x,I2) \/ divset(R_(T2.n),x,R_x,I2)
    by A1,A3,XBOOLE_1:1,Th15,A5
    .=(T2.(n+1))`2 by Def4;
    hence thesis by A6,A7,XTUPLE_0:2;
  end;
A8: P[n] from NAT_1:sch 2(A2,A4);
A9:dom T1 = NAT = dom T2 by Def4;
  then o in dom T1 implies T1.o=T2.o by A8;
  hence thesis by A9,FUNCT_1:2;
end;
