reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th97:
  for r be Sequence of REAL
    for y be Surreal-Sequence
      for s be uSurreal-Sequence,alpha be Ordinal st
         s,y,r simplest_up_to alpha & s|alpha is one-to-one
  holds (born s)|alpha is increasing
proof
  let r be Sequence of REAL;
  let y be Surreal-Sequence;
  let s be uSurreal-Sequence;
  let alpha be Ordinal such that
A1:  s,y,r simplest_up_to alpha & s|alpha is one-to-one;
  for A,B be Ordinal st A in B in dom ((born s)|alpha) holds
  ((born s)|alpha).A in ((born s)|alpha).B
  proof
    set bs = born s;
    let A,B be Ordinal such that
A2: A in B in dom (bs|alpha);
A3: B in alpha & B in dom bs = dom s by Def20,A2,RELAT_1:57;
A4: s,y,r simplest_on_position B by A1,A2;
A5: B in dom s & A in dom s by A2,A3,ORDINAL1:10;
    then s.B in rng s & s.A in rng s by FUNCT_1:def 3;
    then reconsider sB=s.B,sA=s.A as uSurreal by SURREALO:def 12;
A6: A in alpha by A2,ORDINAL1:10;
A7: s,y,r simplest_on_position A by A1,A2,ORDINAL1:10;
A8: A c= B by A2,ORDINAL1:def 2;
    sB in_meets_terms s,y,r,B by A4;
    then
A9: sB in_meets_terms s,y,r,A by A8;
A10: A<>B by A2;
    B in dom (s|alpha) & A in dom (s|alpha) & sA = (s|alpha).A &
    sB=(s|alpha).B by RELAT_1:57,A2,A5,A6,FUNCT_1:49;
    then
A11: sA<>sB by A10,A1,FUNCT_1:def 4;
    (bs|alpha).A = bs.A & (bs|alpha).B = bs.B
      by A2,ORDINAL1:10,FUNCT_1:49;
    then
A12: (bs|alpha).A = born sA & (bs|alpha).B = born sB
    by Def20,A2,A3,ORDINAL1:10;
    per cases;
    suppose A=0;
      then
A13:  sA=0_No by A7;
      then
A14:  born sA =0 by SURREAL0:37;
A15:  {} c= born sB;
      born sB<>0 by A13,A11,SURREAL0:37;
      hence thesis by A12,A14,ORDINAL1:11,A15,XBOOLE_0:def 8;
    end;
    suppose A<>0;
      hence thesis by A12,A7,A9,A11;
    end;
  end;
  hence thesis by ORDINAL2:def 12;
end;
