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 Th30:
  uDyadic.d in born_eq_set uDyadic.d
proof
A1: - -d = d;
  per cases;
  suppose d is Integer;
    then reconsider i=d as Integer;
A2: uDyadic.d = uInt.i by Def5;
    born uInt.i = born_eq uInt.i by SURREALO:48;
    then uInt.i in Day born_eq uInt.i by SURREAL0:def 18;
    hence thesis by A2,SURREALO:def 6;
  end;
  suppose
A3:   d < 0 & not d is Integer;
      then 0 < -d  & not -d is Integer by A1;
      then consider n,m,p be Nat such that
A4:   - d = n + (2*m+1)/(2|^(p+1)) & 2*m+1 < 2|^(p+1) by Th28;
      set D = uDyadic.-d;
A5:   0_No <= D by A3,Th29;
A6:   born D = n+(p+1)+1 by A4,Th26;
A7:   D = - uDyadic.d by Th27;
      then
A8:   born D = born uDyadic.d by SURREALR:12;
      for y be Surreal st y == D holds born D c= born y
      proof
        let y be Surreal such that
A9:     y == D & not born D c= born y;
A10:    born y in Segm (n+(p+1)+1) by A6,A9,ORDINAL1:16;
        reconsider By = born y as Nat by A6,A9;
        By < n+(p+1)+1 by A10,NAT_1:44;
        then By <= n+p+1 by NAT_1:13;
        then Segm By c= Segm (n+p+1) by NAT_1:39;
        then
A11:    y in Day By c= Day (n+p+1) by SURREAL0:def 18,35;
        not y == uDyadic.(n+p+1)
        proof
          assume y == uDyadic.(n+p+1);
          then -d <= n+p+1 <= -d by Th24,A9,SURREALO:4;
          then -d = n+(p+1) by XXREAL_0:1;
          hence thesis by NAT_1:14,A4, XREAL_1:189;
        end;
        then consider x1,y1,p1 be Nat such that
A12:    y == uDyadic.(x1 + y1 / (2|^p1)) & y1 < 2|^p1 & x1+p1 < n+p+1
        by A5,A9,SURREALO:4,A11,Th25;
        -d <= x1 + y1 / (2|^p1) <= -d by Th24,A9,A12,SURREALO:4;
        then
A13:    n + (2*m+1) / (2|^(p+1)) = x1 + y1 / (2|^p1) by A4,XXREAL_0:1;
        0 <= (2*m+1) / (2|^(p+1)) < 1 & 0<= y1 / (2|^p1) < 1
        by A4,A12,XREAL_1:191;
        then n+0 <= n+ (2*m+1) / (2|^(p+1)) < n+1 & x1+0<= x1+ y1 / (2|^p1)
        < x1+1 by XREAL_1:6;
        then n < x1+1 & x1 < n+1 by A13,XXREAL_0:2;
        then n<= x1 <= n by NAT_1:13;
        then
A14:    x1=n by XXREAL_0:1;
        x1+p1 < n+(p+1) by A12;
        then p1 < p+1 by A14,XREAL_1:6;
        then p1<=p by NAT_1:13;
        then reconsider P=p-p1 as Nat by NAT_1:21;
        p = p1+P;
        then
A15:    2|^p = (2|^p1)*(2|^P) & 2|^(p+1) = 2*(2|^p) by NEWTON:6,8;
        (2*m+1) *(2|^p1) = y1 *(2|^(p+1)) by A13,A14,XCMPLX_1:95
        .= (y1* (2|^P) *2) *(2|^p1) by A15;
        hence thesis by XCMPLX_1:5;
      end;
      then born_eq uDyadic.d = born_eq D = born D
      by A7,SURREALO:def 5,SURREALR:13;
      then uDyadic.d in Day born_eq uDyadic.d by A8,SURREAL0:def 18;
      hence thesis by SURREALO:def 6;
    end;
    suppose
A16:  0 <= d & not d is Integer;
      then consider n,m,p be Nat such that
A17:  d = n + (2*m+1)/(2|^(p+1)) & 2*m+1 < 2|^(p+1) by Th28;
      set D = uDyadic.d;
A18:  0_No <= D by A16,Th29;
A19:  born D = n+(p+1)+1 by A17,Th26;
      for y be Surreal st y == D holds born D c= born y
      proof
        let y be Surreal such that
A20:    y == D & not born D c= born y;
A21:    born y in Segm (n+(p+1)+1) by A19,A20,ORDINAL1:16;
        reconsider By = born y as Nat by A19,A20;
        By < n+(p+1)+1 by A21,NAT_1:44;
        then By <= n+p+1 by NAT_1:13;
        then Segm By c= Segm (n+p+1) by NAT_1:39;
        then
A22:    y in Day By c= Day (n+p+1) by SURREAL0:def 18,35;
        not y == uDyadic.(n+p+1)
        proof
          assume y == uDyadic.(n+p+1);
          then d <= n+p+1 <= d by Th24,A20,SURREALO:4;
          then d = n+(p+1) by XXREAL_0:1;
          hence thesis by NAT_1:14,A17, XREAL_1:189;
        end;
        then consider x1,y1,p1 be Nat such that
A23:    y == uDyadic.(x1 + y1 / (2|^p1)) & y1 < 2|^p1 & x1+p1 < n+p+1
        by A18,A20,SURREALO:4,A22,Th25;
        d <= x1 + y1 / (2|^p1) <= d by Th24,A20,A23,SURREALO:4;
        then
A24:    n + (2*m+1) / (2|^(p+1)) = x1 + y1 / (2|^p1) by A17,XXREAL_0:1;
        0 <= (2*m+1) / (2|^(p+1)) < 1 & 0<= y1 / (2|^p1) < 1
        by A17,A23,XREAL_1:191;
        then n+0 <= n+ (2*m+1) / (2|^(p+1)) < n+1 & x1+0<= x1+ y1 / (2|^p1)
        < x1+1 by XREAL_1:6;
        then n < x1+1 & x1 < n+1 by A24,XXREAL_0:2;
        then n <= x1 <= n by NAT_1:13;
        then
A25:    x1=n by XXREAL_0:1;
        x1+p1 < n+(p+1) by A23;
        then p1 < p+1 by A25,XREAL_1:6;
        then p1<=p by NAT_1:13;
        then reconsider P=p-p1 as Nat by NAT_1:21;
        p = p1+P;
        then
A26:    2|^p = (2|^p1)*(2|^P) & 2|^(p+1) = 2*(2|^p) by NEWTON:6,8;
        (2*m+1) *(2|^p1) = y1 *(2|^(p+1)) by A24,A25,XCMPLX_1:95
        .= (y1* (2|^P) *2) *(2|^p1) by A26;
        hence thesis by XCMPLX_1:5;
      end;
      then born_eq D = born D by SURREALO:def 5;
      then D in Day born_eq D by SURREAL0:def 18;
      hence thesis by SURREALO:def 6;
    end;
end;
