reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;
reserve r,r1,r2 for Real;

theorem Th46:
  sReal.d == uDyadic.d = uReal.d
proof
  set Rd=sReal.d,Dd=uDyadic.d;
  consider i be Integer, k be Nat such that
A1: d = i / (2|^k) by Th18;
A2: L_Rd << {Dd} << R_Rd
  proof
    thus L_Rd << {Dd}
    proof
      let l,r be Surreal such that
A3:   l in L_Rd & r in {Dd};
      consider n such that
A4:   l = uDyadic.([/ d*(2|^n)-1 \] / (2|^n)) by A3,Th42;
      l < Dd by Th41,A4,Th24;
      hence thesis by A3,TARSKI:def 1;
    end;
    let l,r be Surreal such that
A5: l in {Dd} & r in R_Rd;
    consider n such that
A6: r = uDyadic.([\ d*(2|^n)+1 /] / (2|^n)) by A5,Th43;
    Dd < r by Th41,A6,Th24;
    hence thesis by A5,TARSKI:def 1;
  end;
  for z st L_Rd << {z} << R_Rd holds born Dd c= born z
  proof
    born Dd is finite by Th37;
    then
    reconsider B=born Dd as Nat;
    let z such that
A7: L_Rd << {z} << R_Rd;
    assume
A8: not born Dd c= born z;
    then born z in Segm B by ORDINAL1:16;
    then reconsider Z=born z as Nat;
    z in Day Z by SURREAL0:def 18;
    then consider f be Dyadic such that
A9: z == uDyadic.f & uDyadic.f in Day Z by Th35;
    consider j be Integer, m be Nat such that
A10:f = j / (2|^m) by Th18;
    set F = uDyadic.f;
A11:L_Rd << {F} & F in {F} & {F} << R_Rd by A7,A9,SURREALO:17,18,TARSKI:def 1;
    m+k+1 = k+(m+1);
    then 2|^(m+k+1) = (2|^k)*(2|^(m+1)) by NEWTON:8
    .= (2|^k)*(2*2|^m)by NEWTON:6;
    then
A12: d*(2|^(m+k+1)) = d*(2|^k)*(2*2|^m)
    .= i *(2*2|^m) by A1,XCMPLX_1:87;
    per cases by XXREAL_0:1;
    suppose f < d;
      then f < (j*(2|^k)*2+1)/(2|^(m+k+1))
      <=(i*(2|^m)*2-1)/(2|^(m+k+1)) by Th45,A10,A1;
      then
A13:  f < (i*(2|^m)*2-1)/(2|^(m+k+1)) by XXREAL_0:2;
      [/ d*(2|^(m+k+1))-1 \] = i *(2*2|^m)-1 by A12;
      then uDyadic.((i*(2|^m)*2-1)/(2|^(m+k+1))) <= F by A11,Th42;
      hence thesis by A13,Th24;
    end;
    suppose d < f;
      then (i*(2|^m)*2+1)/(2|^(m+k+1))
      <=(j*(2|^k)*2-1)/(2|^(m+k+1)) < f by Th45,A10,A1;
      then
A14:  (i*(2|^m)*2+1)/(2|^(m+k+1))< f by XXREAL_0:2;
      [\ d*(2|^(m+k+1))+1 /] = i *(2*2|^m)+1 by A12;
      then F<= uDyadic.((i*(2|^m)*2+1)/(2|^(m+k+1))) by A11,Th43;
      hence thesis by A14,Th24;
    end;
    suppose d = f;
      hence thesis by A9,SURREAL0:def 18,A8;
    end;
  end;
  hence
A15:sReal.d == uDyadic.d by A2,SURREALO:16;
  uReal.d = Unique_No sReal.d by Def7;
  then sReal.d == uReal.d by SURREALO:def 10;
  then uReal.d == uDyadic.d by A15,SURREALO:4;
  hence thesis by SURREALO:50;
end;
