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;

theorem Th26:
  2*m+1 < 2|^p implies born uDyadic.(n + (2*m+1) / (2|^p)) = n+p+1
proof
  set d = n + (2*m+1) / (2|^p);
  assume
A1:2*m+1 < 2|^p & born uDyadic.d <> n+p+1;
A2: not uDyadic.d == uDyadic.(n+p)
  proof
    assume uDyadic.d == uDyadic.(n+p);
    then d <= n+p <= d by Th24;
    then d = n+p by XXREAL_0:1;
    then p * (2|^p) < 1 * (2|^p) by A1,XCMPLX_1:87;
    then p=0 by NAT_1:14,XREAL_1:64;
    then 2|^p =1+0 by NEWTON:4;
    hence thesis by A1,NAT_1:13;
  end;
  n+p < n+p+1 by NAT_1:13;
  then
A3:0_No <= uDyadic.d in Day (n+p+1) by A1,Th25;
  for O st uDyadic.d in Day O holds (n+p+1) c= O
  proof
    let O such that
A4: uDyadic.d in Day O and
A5: not (n+p+1) c= O;
A6: O in n+p+1 = Segm (n+p+1) by A5,ORDINAL1:16;
    reconsider O as Nat by A5;
    O < n+p+1 by A6,NAT_1:44;
    then Segm O = O <= n+p= Segm (n+p) by NAT_1:13;
    then Day O c= Day (n+p) by NAT_1:39,SURREAL0:35;
    then consider x,y,p1 be Nat such that
A7: uDyadic.d == uDyadic.(x + y / (2|^p1)) &
    y < 2|^p1 & x+p1 < n+p by A4,Th25,A2,A3;
    d <= x + y / (2|^p1) <= d by A7,Th24;
    then
A8: n + (2*m+1) / (2|^p) = x + y / (2|^p1) by XXREAL_0:1;
    0 <= (2*m+1) / (2|^p) < 1 & 0<= y / (2|^p1) < 1 by A1,A7,XREAL_1:191;
    then n+0 <= n+ (2*m+1) / (2|^p) < n+1 & x+0<= x+ y / (2|^p1) < x+1
    by XREAL_1:6;
    then n < x+1 & x < n+1 by A8,XXREAL_0:2;
    then n<= x <= n by NAT_1:13;
    then
A9: x=n by XXREAL_0:1;
    then p1 < p by A7,XREAL_1:6;
    then p1+1<=p by NAT_1:13;
    then reconsider P=p-(p1+1) as Nat by NAT_1:21;
    p = p1+1+P;
    then
A10:2|^p = (2|^(p1+1))*(2|^P) & 2|^(p1+1) = 2*(2|^p1)
    by NEWTON:6,8;
    (2*m+1) *(2|^p1) = y *(2|^p) by A8,A9,XCMPLX_1:95
    .= (y* (2|^P) *2) *(2|^p1) by A10;
    hence thesis by XCMPLX_1:5;
  end;
  hence thesis by A1,SURREAL0:def 18,A3;
end;
