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

theorem Th77:
for r be Sequence of REAL
  for y be Sequence
       for s1,s2 be uSurreal-Sequence
           for alpha be Ordinal st
       alpha c= dom s1 & alpha c= dom s2 &
      s1,y,r simplest_up_to alpha &
      s2,y,r simplest_up_to alpha
    holds s1|alpha = s2|alpha
proof
  let r be Sequence of REAL;
  let y be Sequence;
  let s1,s2 be uSurreal-Sequence;
  let alpha be Ordinal such that
A1:alpha c= dom s1 & alpha c= dom s2 and
A2:s1,y,r simplest_up_to alpha & s2,y,r simplest_up_to alpha;
  defpred P[Ordinal]
  means $1 in alpha implies s1.$1 = s2.$1;
A3: for D be Ordinal st for C be Ordinal st C in D holds P[C] holds P[D]
  proof
    let D be Ordinal such that
A4: for C be Ordinal st C in D holds P[C];
    assume
A5: D in alpha;
    then
A6: s1,y,r simplest_on_position D &
    s2,y,r simplest_on_position D by A2;
A7: D c= alpha by A5,ORDINAL1:def 2;
A8: dom (s1|D)=D = dom (s2|D) by RELAT_1:62,A5,ORDINAL1:def 2,A1;
A9: for x be object st x in D holds (s1|D).x = (s2|D).x
    proof
      let x be object such that
A10:  x in D;
      reconsider o=x as Ordinal by A10;
      thus (s1|D).x = s1.o by A10,FUNCT_1:49
      .= s2.o by A7,A4,A10
      .= (s2|D).x by A10,FUNCT_1:49;
    end;
    s1.D in rng s1 & s2.D in rng s2 by A1,A5,FUNCT_1:def 3;
    then s1.D is uSurreal & s2.D is uSurreal by SURREALO:def 12;
    hence thesis by A9,A6,A8,FUNCT_1:2,Th76;
  end;
A11:for D be Ordinal holds P[D] from ORDINAL1:sch 2(A3);
A12:dom (s1|alpha)=alpha = dom (s2|alpha) by A1,RELAT_1:62;
  for x be object st x in alpha holds (s1|alpha).x = (s2|alpha).x
  proof
    let x be object such that
A13:x in alpha;
    reconsider o=x as Ordinal by A13;
    thus (s1|alpha).x = s1.o by A13,FUNCT_1:49
    .= s2.o by A11,A13
    .= (s2|alpha).x by A13,FUNCT_1:49;
  end;
  hence thesis by A12,FUNCT_1:2;
end;
