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 Th25:
  (0_No <= z & z in Day n & not z == uDyadic.n implies
     ex x,y,p be Nat st
        z == uDyadic.(x + y / (2|^p)) & y < 2|^p & x+p < n) &
  for x,y,p be Nat st y < 2|^p & x+p < n holds
      0_No <= uDyadic.(x + y / (2|^p)) in Day n
proof
  defpred P[Nat] means
   (for s be Surreal st s in Day $1 & 0_No <= s holds
          s == uDyadic.$1 or ex d be Dyadic,x,y,p be Nat st
                  s == uDyadic.d & y < 2|^p &
                  d = x + y / (2|^p) & x+p < $1) &
     (for x,y,p be Nat st y < 2|^p &
                   x+p < $1 holds 0_No <= uDyadic.(x + y / (2|^p)) in Day $1);
A1:P[0]
  proof
A2: 0_No = uInt.0 = uDyadic.0 by Def1,Def5;
    thus for s be Surreal st s in Day 0 & 0_No <= s holds s == uDyadic.0 or
    ex d be Dyadic,x,y,p be Nat st s == uDyadic.d & y < 2|^p &
    d = x + y / (2|^p) & x+p < 0 by A2,SURREAL0:2,TARSKI:def 1;
    let x,y,p be Nat such that
A3: y < 2|^p & x+p < 0;
    thus 0_No <= uDyadic.(x + y / (2|^p)) in Day 0 by A3;
  end;
A4:for n holds P[n] implies P[n+1]
  proof
    let n;
    assume
A5: P[n];
    set n1=n+1;
A6: 0_No = uInt.0 = uDyadic.0 by Def1,Def5;
    n < n+1 by NAT_1:13;
    then
A7: Segm n c= Segm (n+1) & n in Segm (n+1) by NAT_1:39,44;
    then
A8: Day n c= Day (n+1) by SURREAL0:35;
    thus for s be Surreal st s in Day n1 & 0_No <= s holds
    s == uDyadic.n1 or ex d be Dyadic,x,y,p be Nat st
    s == uDyadic.d & y < 2|^p &
    d = x + y / (2|^p) & x+p < n1
    proof
      let s be Surreal such that
A9:   s in Day n1 & 0_No <= s;
      set c = Unique_No s;
A10:  c == s by SURREALO:def 10;
      then
A11:  0_No <= c by A9,SURREALO:4;
      per cases;
      suppose c in Day n;
        then per cases by A10,A9,SURREALO:4,A5;
        suppose
A12:      c == uDyadic.n;
          ex d be Dyadic,x,y,p be Nat st
          s == uDyadic.d & y < 2|^p & d = x + y / (2|^p) & x+p < n1
          proof
            take d = n, x= n,y=0,p=0;
            thus thesis by A12,A10,SURREALO:4,NAT_1:13;
          end;
          hence thesis;
        end;
        suppose ex d be Dyadic,x,y,p be Nat st
          c == uDyadic.d & y < 2|^p &
          d = x + y / (2|^p) & x+p < n;
          then consider d be Dyadic,x,y,p be Nat such that
A13:      c == uDyadic.d & y < 2|^p & d = x + y / (2|^p) & x+p < n;
          x+p < n1 & s == uDyadic.d by SURREALO:4,A10,A13,NAT_1:13;
          hence thesis by A13;
        end;
      end;
      suppose
A14:    not c in Day n;
A15:    born_eq c = born c by SURREALO:48;
A16:    born_eq c = born_eq s c= born s c= n1
          by A9,SURREAL0:def 18,SURREALO:def 5,33,SURREALO:def 10;
        then
A17:    born_eq c c= n1 by XBOOLE_1:1;
A18:    c in Day born c by SURREAL0:def 18;
A19:    n1 c= born_eq c
        proof
          assume not n1 c= born_eq c;
          then born_eq c in Segm n1 = succ (Segm n) by ORDINAL1:16,NAT_1:38;
          then Day born c c= Day n by SURREAL0:35,A15,ORDINAL1:22;
          hence thesis by A14,A18;
        end;
        then
A20:    n1 = born c by A15,A17,XBOOLE_0:def 10;
        assume
A21:    not s == uDyadic.n1;
        per cases;
        suppose
A22:      c = 0_No;
          take d=0,x=0,y=0,p=0;
          thus thesis by A22,SURREALO:def 10,A6;
        end;
        suppose
A23:      c <> 0_No;
          R_c <>{}
          proof
            assume
A24:        R_ c ={};
            then L_ c <>{} by A23;
            then card (L_c) = 1 by SURREALO:8,44,A15,A16;
            then consider y be object such that
A25:        L_c = {y} by CARD_2:42;
            y in L_c by A25,TARSKI:def 1;
            then reconsider y as Surreal by SURREAL0:def 16;
A26:        not c == 0_No by A23,SURREALO:50;
            c = [{y},{}] by A24,A25;
            then consider k be Nat such that
A27:        c == uInt.(k+1) & uInt.k <= y < uInt.(k+1) & k in born c
            by A26,Th16,A15,A16,Th17;
            c = uInt.(k+1) by A27,SURREALO:50;
            then c = uInt.n1 = uDyadic.n1 by A20,Th4,Def5;
            hence contradiction by SURREALO:def 10,A21;
          end;
          then card (R_c) = 1 by A15,A16,SURREALO:8,45;
          then consider c2 be object such that
A28:      R_c = {c2} by CARD_2:42;
          c2 in R_c by A28,TARSKI:def 1;
          then reconsider c2 as Surreal by SURREAL0:def 16;
          L_ c <>{}
          proof
            assume
A29:        L_c ={};
            -c = [--R_c, --L_c] by SURREALR:7;
            then
A30:        -c = [{-c2},{}] by A29,A28,SURREALR:21,22;
            not c == 0_No by A23,SURREALO:50;
            then not -c == 0_No by SURREALR:23,10;
            then c2 <= 0_No by A30,Th16,SURREALR:23,10;
            then not (L_0_No << {c} & {0_No} << R_c) by A28,SURREALO:21;
            hence thesis by A11,SURREAL0:43;
          end;
          then card (L_c) = 1 by A15,A16,SURREALO:8,44;
          then consider c1 be object such that
A31:      L_c = {c1} by CARD_2:42;
          c1 in L_c by A31,TARSKI:def 1;
          then reconsider c1 as Surreal by SURREAL0:def 16;
A32:      c = [{c1},{c2}] by A28,A31;
          not c == 0_No by A23,SURREALO:50;
          then
A33:      not (L_c << {0_No} & {c} << R_0_No)
          by A10,A9,SURREAL0:43,SURREALO:4;
          then 0_No <= c1 by A31,SURREALO:21;
          then
A34:      0_No <= c2 by A32,SURREALO:4,22;
          c1 in L_c & c2 in R_c by A28,A31,TARSKI:def 1;
          then c1 in L_c \/ R_c & c2 in L_c \/ R_c by XBOOLE_0:def 3;
          then
A35:      born c1 in n1 & born c2 in n1 & n1= Segm n1
          by A20,SURREALO:1;
          then reconsider b1=born c1, b2=born c2 as Nat;
          b1 < n1 & b2 < n1 by A35,NAT_1:44;
          then b1 <= n & b2 <= n & n = Segm n by NAT_1:13;
          then Segm b1 c= n & Segm b2 c= n by NAT_1:39;
          then
A36:      c1 in Day b1 c= Day n & c2 in Day b2 c= Day n
          by SURREAL0:35,SURREAL0:def 18;
A37:      uDyadic.n = uInt.n by Def5;
          not c1 == uDyadic.n
          proof
            assume c1 == uDyadic.n;
            then uInt.n < c2 by A37,A32,SURREALO:4,22;
            hence thesis by Th2,A36;
          end;
          then consider d1 be Dyadic,x1,y1,p1 be Nat such that
A38:      c1 == uDyadic.d1 & y1 < 2|^p1 &
          d1 = x1 + y1 / (2|^p1) & x1+p1 < n by A36,A33,A31,SURREALO:21,A5;
          y1 / (2|^p1) < 1 by A38,XREAL_1:189;
          then
A39:      d1 < x1+1 by A38,XREAL_1:6;
A40:      born c = born_eq c by SURREALO:48;
          per cases by A5,A34,A36;
          suppose
A41:        c2 == uDyadic.n;
A42:        x1+1 = n
            proof
              assume
A43:          x1+1 <>n;
              x1 <= x1+p1 by NAT_1:11;
              then x1 < n by A38,XXREAL_0:2;
              then x1+1 <= n by NAT_1:13;
              then x1+1 < n by A43,XXREAL_0:1;
              then
A44:          2 <= x1+2=x1+1+1 <= n by NAT_1:11,13;
              then reconsider N2=n-2 as Nat by NAT_1:21,XXREAL_0:2;
              x1+2 <= N2+2 by A44;
              then x1 <= N2 by XREAL_1:6;
              then x1+1 <= N2+1 by XREAL_1:6;
              then d1 < N2+1 by A39,XXREAL_0:2;
              then c1 < uDyadic.(N2+1) by A38,SURREALO:4,Th24;
              then
A45:          {c1} << {uDyadic.(N2+1)} by SURREALO:21;
              N2+1 < N2+1+1 by NAT_1:13;
              then uDyadic.(N2+1) < c2 by A41,SURREALO:4,Th24;
              then {uDyadic.(N2+1)} << {c2} by SURREALO:21;
              then
A46:          born c c= born uDyadic.(N2+1)
              by A45, A28,A31,A40,SURREALO:51;
              uDyadic.(N2+1) = uInt.(N2+1) by Def5;
              then Segm n1 c= Segm (N2+1) by A46,A20,Th4;
              then n1 <= N2+1 by NAT_1:39;
              then n1 <= N2+1+1 by NAT_1:13;
              hence thesis by NAT_1:13;
            end;
            x1+p1+1 <= n by A38,NAT_1:13;
            then n+p1 <= n+0 by A42;
            then
A47:        p1=0 & 2|^0=1 by XREAL_1:6,NEWTON:4;
            then c1 == uDyadic.x1 by A38,NAT_1:14;
            then
A48:        {c1} <==> {uDyadic.x1} by SURREALO:32;
A49:        {c2} <==> {uDyadic.(x1+1)} by A41,A42,SURREALO:32;
            uDyadic.x1 = uDyadic.(x1/(2|^0)) &
            uDyadic.(x1+1) = uDyadic.((x1+1)/(2|^0)) by A47;
            then uDyadic.((2*x1+1)/(2|^(0+1)))=[{uDyadic.x1},{uDyadic.(x1+1)}]
            by Def5;
            then
            c == uDyadic.((2*x1+1)/(2|^(0+1))) by SURREALO:29,A48,A49,A32;
            then
A50:        s == uDyadic.((2*x1+1)/(2|^1)) by SURREALO:4,A10;
A51:        (2*x1+1)/(2|^1) = x1+ 1/(2|^1);
            x1 + 1 <n1 by A42,NAT_1:13;
            hence thesis by A50,A51;
          end;
          suppose ex d be Dyadic,x,y,p be Nat st
            c2 == uDyadic.d & y < 2|^p &
            d = x + y / (2|^p) & x+p < n;
            then consider d2 be Dyadic,x2,y2,p2 be Nat such that
A52:        c2 == uDyadic.d2 & y2 < 2|^p2 &
            d2 = x2 + y2 / (2|^p2) & x2+p2 < n;
            c1 < uDyadic.d2 by A32,SURREALO:4,22,A52;
            then
A53:        uDyadic.d1 < uDyadic.d2 by A38,SURREALO:4;
            y2 / (2|^p2) < 1 by A52,XREAL_1:189;
            then
A54:        d2 < x2+1 by A52,XREAL_1:6;
A55:        x1+0 <= d1 by A38,XREAL_1:6;
            per cases;
            suppose
A56:          x1 <> x2;
              x1 < x2
              proof
                assume x2 <= x1;
                then x2 < x1 by A56,XXREAL_0:1;
                then x2 +1 <= x1 by NAT_1:13;
                then d2 < x1 by A54,XXREAL_0:2;
                hence thesis by A53,Th24,A55,XXREAL_0:2;
              end;
              then
A57:          x1 +1 <= x2 by NAT_1:13;
              x2+0 <= d2 by A52,XREAL_1:6;
              then
A58:          x1+1 <= d2 by A57,XXREAL_0:2;
A59:          d2 = x1+1
              proof
                assume d2 <> x1+1;
                then x1+1 < d2 by A58,XXREAL_0:1;
                then c1 < uDyadic.(x1+1) < c2 by A38,SURREALO:4,A52,A39,Th24;
                then {c1} << {uDyadic.(x1+1)} << {c2} by SURREALO:21;
                then born c c= born uDyadic.(x1+1) = born uInt.(x1+1)=x1+1
                by A40,SURREALO:51,A28,A31,Def5,Th4;
                then Segm n1 c= Segm (x1+1) by A19,A15,XBOOLE_1:1;
                then n1 <= x1+1 by NAT_1:39;
                then n1 <= x2 <= x2+p2 by NAT_1:11,A57,XXREAL_0:2;
                then n1 <= x2+p2 by XXREAL_0:2;
                hence thesis by NAT_1:13,A52;
              end;
              y1+1 = 2|^p1
              proof
A60:            y1+1 <= 2|^p1 by A38,NAT_1:13;
                assume y1+1 <> 2|^p1;
                then
A61:            y1 < y1+1 < 2|^p1 by NAT_1:13,A60,XXREAL_0:1;
                then y1/(2|^p1) < (y1+1)/(2|^p1) < (2|^p1)/(2|^p1) =1
                by XREAL_1:74,XCMPLX_1:60;
                then d1 < x1+(y1+1)/(2|^p1) < x1+1 by A38,XREAL_1:6;
                then d1 < x1+(y1+1)/(2|^p1) < d2 by A58,XXREAL_0:2;
                then c1 < uDyadic.(x1+(y1+1)/(2|^p1)) < c2
                by Th24,A52,A38,SURREALO:4;
                then {c1} << {uDyadic.(x1+(y1+1)/(2|^p1))} << {c2}
                by SURREALO:21;
                then
A62:            n1 c= born uDyadic.(x1+(y1+1)/(2|^p1))
                by A40,SURREALO:51,A20,A28,A31;
                uDyadic.(x1+(y1+1)/(2|^p1)) in Day n by A38,A5,A61;
                then born uDyadic.(x1+(y1+1)/(2|^p1)) c= n by SURREAL0:def 18;
                then Segm n1 c= Segm n by XBOOLE_1:1,A62;
                then n1 <=n by NAT_1:39;
                hence thesis by NAT_1:13;
              end;
              then
A63:          ((x1* (2|^p1) + y1+1) / (2|^p1)) = ((x1+1)* (2|^p1)) / (2|^p1)
              .= (x1+1)* ((2|^p1)/(2|^p1)) by XCMPLX_1:74
              .= d2 by A59,XCMPLX_1:88;
A64:          d1 = (x1* (2|^p1) + y1) / (2|^p1) by A38,XCMPLX_1:113;
A65:          {c1} <==> {uDyadic.d1} & {c2} <==> {uDyadic.d2}
              by A52,A38,SURREALO:32;
              uDyadic.(((x1* (2|^p1) + y1)*2+1)/(2|^(p1+1))) =
              [{uDyadic.(d1)},{uDyadic.d2}] by Def5,A64,A63;
              then c == uDyadic.(((x1* (2|^p1) + y1)*2+1)/(2|^(p1+1)))
              by A65,SURREALO:29,A32;
              then
A66:          s == uDyadic.(((x1* (2|^p1) + y1)*2+1)/(2|^(p1+1)))
              by A10,SURREALO:4;
A67:          2|^(p1+1) = 2 * (2|^p1) by NEWTON:6;
              then
A68:          ((x1* (2|^p1) + y1)*2+1)/(2|^(p1+1))
              = ((x1* (2|^(p1+1))) + (y1*2+1))/(2|^(p1+1))
              .= x1 + (y1*2+1)/(2|^(p1+1)) by XCMPLX_1:113;
              y1+1 <= 2|^p1 by A38,NAT_1:13;
              then 2*y1+1+1=2*(y1+1) <= 2*2|^p1 by XREAL_1:64;
              then
A69:          2*y1+1 < 2|^(p1+1) by A67,NAT_1:13;
              x1+p1+1 < n1 by A38,XREAL_1:6;
              then x1+(p1+1) < n1;
              hence thesis by A66,A68,A69;
            end;
            suppose
A70:          x1=x2;
              then
A71:          y1 / (2|^p1) < y2 / (2|^p2) by A38,A52,A53,Th24,XREAL_1:6;
              ex Y1,Y2,p3 be Nat st Y1 < Y2 & Y1 < 2|^p3 & Y2 < 2|^p3 &
              d1 = x1 + Y1 / (2|^p3) & d2 = x2 + Y2 / (2|^p3) & x2+p3 < n
              proof
                per cases;
                suppose p2 < p1;
                  then reconsider p = p1-p2 as Nat by NAT_1:21;
                  take y1,y3=y2*(2|^p),p3=p1;
A72:              2|^(p2+p) = (2|^p2) * (2|^p) by NEWTON:8;
                  y2/(2|^p2) = y3/(2|^p3) by A72,XCMPLX_1:91;
                  hence thesis
                  by A70,A71,XREAL_1:72,A72,A52,XREAL_1:68,A38;
                end;
                suppose p1 <= p2;
                  then reconsider p = p2-p1 as Nat by NAT_1:21;
                  take y3=y1*(2|^p),y2,p3=p2;
A73:              2|^(p1+p) = (2|^p1) * (2|^p) by NEWTON:8;
                  y1/(2|^p1) = y3/(2|^p3) by A73,XCMPLX_1:91;
                  hence thesis by A73,A38,XREAL_1:68,A71,XREAL_1:72,A52;
                end;
              end;
              then consider Y1,Y2,p3 be Nat such that
A74:          Y1 < Y2 & Y1 < 2|^p3 & Y2 < 2|^p3 &
              d1 = x1 + Y1 / (2|^p3) & d2 = x2 + Y2 / (2|^p3) & x2+p3 < n;
              Y2-Y1 >0 by A74,XREAL_1:50;
              then reconsider y = Y2-Y1 as Nat;
A75:          y >= 1+0 by A74,XREAL_1:50,NAT_1:13;
A76:          y = 1
              proof
                assume y <>1;
                then
A77:            y > 1 by A75,XXREAL_0:1;
                Y1 < Y1 +1 < Y1+y=Y2 by A77,NAT_1:13,XREAL_1:6;
                then Y1 / (2|^p3) < (Y1 +1) / (2|^p3) < Y2 / (2|^p3)
                by XREAL_1:74;
                then uDyadic.d1 < uDyadic.(x1+(Y1 +1) / (2|^p3)) < uDyadic.d2
                by Th24,A74,A70,XREAL_1:6;
                then c1 < uDyadic.(x1+(Y1 +1) / (2|^p3)) < c2
                by A52,A38,SURREALO:4;
                then {c1} << {uDyadic.(x1+(Y1 +1) / (2|^p3))} << {c2}
                by SURREALO:21;
                then
A78:            n1 c= born uDyadic.(x1+(Y1 +1) / (2|^p3))
                by A40,SURREALO:51,A20,A28,A31;
                Y1+1 <= Y2 by A74,NAT_1:13;
                then (Y1 +1)<(2|^p3) by A74,XXREAL_0:2;
                then uDyadic.(x1+(Y1 +1) / (2|^p3)) in Day n by A5,A74,A70;
                then born uDyadic.(x1+(Y1 +1) / (2|^p3)) c= n
                by SURREAL0:def 18;
                then Segm n1 c= Segm n by A78,XBOOLE_1:1;
                then n1 <= n by NAT_1:39;
                hence thesis by NAT_1:13;
              end;
A79:          {c1} <==> {uDyadic.d1} & {c2} <==> {uDyadic.d2}
              by A52,A38,SURREALO:32;
A80:          (x1* (2|^p3) + Y1)/(2|^p3) = d1 by A74,XCMPLX_1:113;
A81:          d2 = (x1* (2|^p3) + Y2)/(2|^p3) by A70,A74,XCMPLX_1:113
              .= (x1* (2|^p3) + Y1+1)/(2|^p3) by A76;
              uDyadic.(((x1* (2|^p3) + Y1)*2+1)/(2|^(p3+1))) =
              [{uDyadic.d1},{uDyadic.d2}]
              by Def5,A80,A81;
              then c == uDyadic.(((x1* (2|^p3) + Y1)*2+1)/(2|^(p3+1)))
              by A79,SURREALO:29,A32;
              then
A82:          s == uDyadic.(((x1* (2|^p3) + Y1)*2+1)/(2|^(p3+1)))
              by A10,SURREALO:4;
A83:          2|^(p3+1) = 2 * (2|^p3) by NEWTON:6;
              then
A84:          ((x1* (2|^p3) + Y1)*2+1)/(2|^(p3+1))
              = ((x1* (2|^(p3+1))) + (Y1*2+1))/(2|^(p3+1))
              .= x1 + (Y1*2+1)/(2|^(p3+1)) by XCMPLX_1:113;
              Y1+1 <= 2|^p3 by A74,NAT_1:13;
              then 2*Y1+1+1=2*(Y1+1) <= 2*2|^p3 by XREAL_1:64;
              then
A85:          2*Y1+1 < 2|^(p3+1) by A83,NAT_1:13;
              x1+p3+1 < n1 by A70,A74,XREAL_1:6;
              then x1+(p3+1) < n1;
              hence thesis by A82,A84,A85;
            end;
          end;
        end;
      end;
    end;
    let x,y,p be Nat such that
A86: y < 2|^p & x+p < n1;
    set d = x + y / (2|^p);
A87:x+p <= n by A86,NAT_1:13;
    per cases;
    suppose
A88:  p=0;
      then 2|^p = 1 by NEWTON:4;
      then y = 0 by A86,NAT_1:14;
      then
A89:  uDyadic.d = uInt.x by Def5;
      0 < x or 0 = x;
      then
A90:  uInt.0 < uInt.x or uInt.0 = uInt.x by Lm4;
      Segm x c= Segm n1 by A88,A86,NAT_1:39;
      then uInt.x in Day x c= Day n1 by Th1,SURREAL0:35;
      hence thesis by A90,Def1,A89;
    end;
    suppose
A91:  y is even & p<>0;
      then consider z be Nat such that
A92:  y = 2*z by ABIAN:def 2;
      reconsider p1=p-1 as Nat by NAT_1:20,A91;
      p=p1+1;
      then
A93:  2|^p = 2*2|^p1 by NEWTON:6;
      then
A94:  y / (2|^p) = z /2|^p1 by A92,XCMPLX_1:91;
A95:  z < 2|^p1 by A93,A92,A86,XREAL_1:64;
      x+p1+1 <= n by A86,NAT_1:13;
      then x+p1 < n by NAT_1:13;
      then 0_No <= uDyadic.(x + z / (2|^p1)) in Day n by A5,A95;
      hence thesis by A8,A94;
    end;
    suppose
A96:  y is odd & p<>0;
      then consider z be Nat such that
A97:  y = 2*z+1 by ABIAN:9;
      reconsider p1=p-1 as Nat by NAT_1:20,A96;
      set SD= uDyadic.((x*2|^p1+z)/(2|^p1)),
      SD1 = uDyadic.(((x*2|^p1+z)+1)/(2|^p1));
      p=p1+1;
      then
A98:  2|^p = 2*2|^p1 by NEWTON:6;
      then
A99:  x*(2|^p) + y = 2*(x*2|^p1+z)+1 by A97;
      d = (2*(x*2|^p1+z)+1) / (2|^(p1+1)) by A99,XCMPLX_1:113;
      then
A100: uDyadic.d = [{SD},{SD1}] by Def5;
A101: z < 2|^p1 by XREAL_1:64,A98,A97,A86,NAT_1:13;
      x+p1+1 <= n by A86,NAT_1:13;
      then
A102: x+p1 < n by NAT_1:13;
      (x*2|^p1+z)/(2|^p1) = x + z/(2|^p1) by XCMPLX_1:113;
      then
A103: SD in Day n by A5,A101,A102;
      2*z+1+1 <= 2*2|^p1 by A98,A97,A86,NAT_1:13;
      then 2*(z+1) <= 2*2|^p1;
      then
A104: z+1 <= 2|^p1 by XREAL_1:68;
A105: SD1 in Day n
      proof
        (x*2|^p1+z)+1 = (x*2|^p1)+(z+1);
        then
A106:   ((x*2|^p1+z)+1)/(2|^p1) = x + (z+1)/(2|^p1) by XCMPLX_1:113;
        per cases by A104,XXREAL_0:1;
        suppose z+1 = 2|^p1;
          then ((x*2|^p1+z)+1)/(2|^p1) = x+1 by A106,XCMPLX_1:60;
          then
A107:     SD1 = uInt.(x+1) by Def5;
          x+1 <= x+1+p1 = x+p by NAT_1:11;
          then Segm (x+1) c= Segm n by NAT_1:39,A87,XXREAL_0:2;
          then uInt.(x+1) in Day (x+1) c= Day n by SURREAL0:35,Th1;
          hence thesis by A107;
        end;
        suppose z+1 < 2|^p1;
          hence thesis by A102,A5,A106;
        end;
      end;
A108: {SD} << {SD1} by SURREALO:21,22,A100;
      for o be object st o in {SD} \/ {SD1} ex O st O in n1 & o in Day O
      proof
        let o be object;
        assume o in {SD} \/ {SD1};
        then o in {SD} or o in {SD1} by XBOOLE_0:def 3;
        then o in Day n by A103,A105,TARSKI:def 1;
        hence thesis by A7;
      end;
      hence thesis by A6,Th24,A100,SURREAL0:46,A108;
    end;
  end;
A109:for n holds P[n] from NAT_1:sch 2(A1,A4);
  thus 0_No <= z & z in Day n & not z == uDyadic.n implies
  ex x,y,p be Nat st z == uDyadic.(x + y / (2|^p)) & y < 2|^p & x+p < n
  proof
    assume 0_No <= z & z in Day n & not z == uDyadic.n;
    then ex d be Dyadic,x,y,p be Nat st
    z == uDyadic.d & y < 2|^p & d = x + y / (2|^p) & x+p < n by A109;
    hence thesis;
  end;
  thus thesis by A109;
end;
