reserve n,m,k for Nat,
        p,q for n-element XFinSequence of NAT,
        i1,i2,i3,i4,i5,i6 for Element of n,
        a,b,c,d,e for Integer;

theorem
  for i1,i2,i3 holds
    {p:p.i2 =(p.i1) |^ (p.i3)} is diophantine Subset of n -xtuples_of NAT
proof
  let i1,i2,i3;
  set n7=n+7;
  set WW = {p:  p.i2 =(p.i1) |^ (p.i3)};
  WW c= n -xtuples_of NAT
  proof
    let y be object;
    assume y in WW;
    then ex p st y=p & p.i2 =(p.i1)|^ (p.i3);
    hence thesis by HILB10_2:def 5;
  end;
  then reconsider WW as Subset of n -xtuples_of NAT;
  per cases;
  suppose n=0;
    then WW is diophantine Subset of n -xtuples_of NAT;
    hence thesis;
  end;
  suppose A1: n<>0;
    n=n+0;then
    reconsider N=n,I1=i1,I2=i2,I3=i3,N1=n+1,N2=n+2,N3=n+3,N4=n+4,N5=n+5,N6=n+6
    as Element of n7 by Th2,Th3;
    defpred P0[XFinSequence of NAT] means $1.I1 = 0;
    A2: {p where p is n7-element XFinSequence of NAT:P0[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th14;
    defpred P1[XFinSequence of NAT] means $1.I1 = 1;
    A3: {p where p is n7-element XFinSequence of NAT:P1[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th14;
    defpred P2[XFinSequence of NAT] means 1*$1.I1 > 0*$1.I2 + 1;
    A4: {p where p is n7-element XFinSequence of NAT:P2[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th7;
    defpred P3[XFinSequence of NAT] means $1.I2 = 0;
    A5: {p where p is n7-element XFinSequence of NAT:P3[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th14;
    defpred P4[XFinSequence of NAT] means $1.I2 = 1;
    A6: {p where p is n7-element XFinSequence of NAT:P4[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th14;
    defpred P5[XFinSequence of NAT] means $1.I3 = 0;
    A7: {p where p is n7-element XFinSequence of NAT:P5[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th14;
    defpred P6[XFinSequence of NAT] means 1*$1.I3 > 0*$1.I1 + 0;
    A8: {p where p is n7-element XFinSequence of NAT:P6[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th7;
    defpred PA[XFinSequence of NAT] means 1*$1.N4 = 1*$1.I3 + 1;
    A9: {p where p is n7-element XFinSequence of NAT:PA[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th6;
    defpred PB[XFinSequence of NAT] means 1*$1.N5 = 1*$1.N3 * $1.I1;
    A10: {p where p is n7-element XFinSequence of NAT:PB[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th9;
    defpred PC[XFinSequence of NAT] means $1.N = Py($1.I1,$1.N4) & $1.I1 >1 ;
    A11: {p where p is n7-element XFinSequence of NAT:PC[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th23;
    defpred PD[XFinSequence of NAT] means 1*$1.N3 > 2 * $1.I3 * $1.N;
    A12: {p where p is n7-element XFinSequence of NAT:PD[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th17;
    defpred PE[XFinSequence of NAT] means $1.N1 = Py($1.N3,$1.N4)
    & $1.N3 > 1;
    A13: {p where p is n7-element XFinSequence of NAT:PE[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th23;
    defpred PF[XFinSequence of NAT] means $1.N2 = Py($1.N5,$1.N4)
    & $1.N5 > 1;
    A14: {p where p is n7-element XFinSequence of NAT:PF[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th23;
    defpred PG[XFinSequence of NAT] means 1*$1.N6 =1* $1.I2 * $1.N1;
    A15: {p where p is n7-element XFinSequence of NAT:PG[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th9;
    defpred PH[XFinSequence of NAT] means 1*$1.N6 >= 1*$1.N2 + 0;
    A16: {p where p is n7-element XFinSequence of NAT:PH[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th8;
    defpred PI[XFinSequence of NAT] means 2*$1.N6 < 1*$1.N1 + 2*$1.N2;
    A17: {p where p is n7-element XFinSequence of NAT:PI[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th18;
    defpred PJ[XFinSequence of NAT] means 1*$1.N2 >= 1*$1.N6 + 0;
    A18: {p where p is n7-element XFinSequence of NAT:PJ[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th8;
    defpred PK[XFinSequence of NAT] means (-2)*$1.N6 < 1*$1.N1 + (-2)*$1.N2;
    A19: {p where p is n7-element XFinSequence of NAT:PK[p]}
    is diophantine Subset of n7 -xtuples_of NAT by Th18;
    defpred P45[XFinSequence of NAT] means P4[$1] & P5[$1];
    defpred P03[XFinSequence of NAT] means P0[$1] & P3[$1];
    defpred P036[XFinSequence of NAT] means P03[$1] & P6[$1];
    defpred P14[XFinSequence of NAT] means P1[$1] & P4[$1];
    defpred P146[XFinSequence of NAT] means P14[$1] & P6[$1];
    defpred P26[XFinSequence of NAT] means P2[$1] & P6[$1];
    defpred PHI[XFinSequence of NAT] means PH[$1] & PI[$1];
    defpred PJK[XFinSequence of NAT] means PJ[$1] & PK[$1];
    {p where p is n7-element XFinSequence of NAT : P4[p] &  P5[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A6,A7);
    then A20: {p where p is n7-element XFinSequence of NAT: P45[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : P0[p] &  P3[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A2,A5);
    then A21: {p where p is n7-element XFinSequence of NAT: P03[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : P03[p] &  P6[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A21,A8);
    then A22: {p where p is n7-element XFinSequence of NAT: P036[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : P1[p] &  P4[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A3,A6);
    then A23: {p where p is n7-element XFinSequence of NAT: P14[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : P14[p] &  P6[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A23,A8);
    then A24: {p where p is n7-element XFinSequence of NAT: P146[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : P2[p] &  P6[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A4,A8);
    then A25: {p where p is n7-element XFinSequence of NAT: P26[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : PH[p] &  PI[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A16,A17);
    then A26: {p where p is n7-element XFinSequence of NAT: PHI[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : PJ[p] &  PK[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A18,A19);
    then A27: {p where p is n7-element XFinSequence of NAT: PJK[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    defpred PHIJK[XFinSequence of NAT] means PHI[$1] or PJK[$1];
    {p where p is n7-element XFinSequence of NAT : PHI[p] or PJK[p]}
    is diophantine Subset of n7 -xtuples_of NAT from UnionDiophantine(A26,A27);
    then A28: {p where p is n7-element XFinSequence of NAT: PHIJK[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    defpred P45036[XFinSequence of NAT] means P45[$1] or P036[$1];
    {p where p is n7-element XFinSequence of NAT : P45[p] or P036[p]}
    is diophantine Subset of n7 -xtuples_of NAT from UnionDiophantine(A20,A22);
    then A29: {p where p is n7-element XFinSequence of NAT: P45036[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    defpred P45036146[XFinSequence of NAT] means P45036[$1] or P146[$1];
    {p where p is n7-element XFinSequence of NAT : P45036[p] or P146[p]}
    is diophantine Subset of n7 -xtuples_of NAT from UnionDiophantine(A29,A24);
    then A30: {p where p is n7-element XFinSequence of NAT: P45036146[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    defpred PAB[XFinSequence of NAT] means PA[$1] & PB[$1];
    defpred PCD[XFinSequence of NAT] means PC[$1] & PD[$1];
    defpred PCDE[XFinSequence of NAT] means PCD[$1] & PE[$1];
    defpred PCDEF[XFinSequence of NAT] means PCDE[$1] & PF[$1];
    defpred PCDEFHIJK[XFinSequence of NAT] means PCDEF[$1] & PHIJK[$1];
    defpred P26CDEFHIJK[XFinSequence of NAT] means P26[$1] & PCDEFHIJK[$1];
    defpred P26CDEFHIJKAB[XFinSequence of NAT] means P26CDEFHIJK[$1] & PAB[$1];
    defpred P26CDEFHIJKABG[XFinSequence of NAT] means P26CDEFHIJKAB[$1] &
      PG[$1];
    {p where p is n7-element XFinSequence of NAT : PA[p] &  PB[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A9,A10);
    then A31: {p where p is n7-element XFinSequence of NAT: PAB[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : PC[p] &  PD[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A11,A12);
    then A32: {p where p is n7-element XFinSequence of NAT: PCD[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : PCD[p] &  PE[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A32,A13);
    then A33: {p where p is n7-element XFinSequence of NAT: PCDE[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : PCDE[p] &  PF[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A33,A14);
    then A34: {p where p is n7-element XFinSequence of NAT: PCDEF[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : PCDEF[p] &  PHIJK[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A34,A28);
    then A35: {p where p is n7-element XFinSequence of NAT: PCDEFHIJK[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : P26[p] &  PCDEFHIJK[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A25,A35);
    then A36:
    {p where p is n7-element XFinSequence of NAT: P26CDEFHIJK[p]}
    is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : P26CDEFHIJK[p] &  PAB[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A36,A31);
    then A37: {p where p is n7-element XFinSequence of NAT:
    P26CDEFHIJKAB[p]} is diophantine Subset of n7 -xtuples_of NAT;
    {p where p is n7-element XFinSequence of NAT : P26CDEFHIJKAB[p] & PG[p]}
    is diophantine Subset of n7 -xtuples_of NAT
      from IntersectionDiophantine(A37,A15);
    then A38: {p where p is n7-element XFinSequence of NAT:
    P26CDEFHIJKABG[p]} is diophantine Subset of n7 -xtuples_of NAT;
    defpred P4503614626CDEFHIJKABG[XFinSequence of NAT] means
    P45036146[$1] or P26CDEFHIJKABG[$1];
    A39: {p where p is n7-element XFinSequence of NAT :
       P45036146[p] or P26CDEFHIJKABG[p]}
       is diophantine Subset of n7 -xtuples_of NAT
    from UnionDiophantine(A30, A38);
    set DD = {p where p is n7-element XFinSequence of NAT:
    P4503614626CDEFHIJKABG[p]};
    set DDR = {p|n where p is n7-element XFinSequence of NAT:p in DD};
    A40: DDR is diophantine Subset of n -xtuples_of NAT by Th5,A39,NAT_1:11;
    A41: DDR c= WW
    proof
      let o be object such that A42: o in DDR;
      consider p be n7-element XFinSequence of NAT such that
      A43: o=p|n & p in DD by A42;
      consider q be n7-element XFinSequence of NAT such that
      A44: p=q & P4503614626CDEFHIJKABG[q] by A43;
      len p = n7 & n7 >=n by CARD_1:def 7,NAT_1:11;
      then len (p|n)=n by AFINSQ_1:54;
      then reconsider pn=p|n as n-element XFinSequence of NAT by CARD_1:def 7;
      set x= pn.i1, y = pn.i2, z = pn.i3;
      set y1 = p.N,y2 = p.N1,y3 = p.N2,K = p.N3;
      A45: x = p.i1 & y = p.i2 & z = p.i3 by A1,Th4;
      (y = 1 & z = 0) or
        (x = 0 & y = 0 & z > 0) or
        (x = 1 & y = 1 & z > 0) or
        (x > 1 & z > 0 & ex y1,y2,y3,K be Nat st
        y1 = Py(x,z+1) & K > 2*z*y1 & y2 = Py(K,z+1) & y3 = Py(K*x,z+1) &
        (0 <= y-y3/y2 <1/2 or 0 <= y3/y2 -y < 1/2))
      proof
        per cases by A44,A45;
        suppose y=1 & z=0;
          hence thesis;
        end;
        suppose x = 0 & y = 0 & z > 0;
          hence thesis;
        end;
        suppose x = 1 & y = 1 & z > 0;
          hence thesis;
        end;
        suppose A46: ((1*x > 0*y + 1) & (1*z > 0*x + 0) & (y1 = Py(x,z+1))
          & (1*K > 2*z*y1)
          & (y2 = Py(K,z+1))
          & (1*y3 = Py(K*x,z+1))
          & (( (1*y*y2 >= 1*y3+0) & (2*y*y2 < 1*y2+2*y3)) or
          ( (1*y3 >= y*y2) & ( (-2)*y*y2 < 1*y2 + (-2)*y3))));
          (x > 1 & z > 0 & ex y1,y2,y3,K be Nat st
          y1 = Py(x,z+1) & K > 2*z*y1 & y2 = Py(K,z+1) & y3 = Py(K*x,z+1) &
          (0 <= y-y3/y2 <1/2 or 0 <= y3/y2 -y < 1/2))
          proof
            thus x > 1 & z > 0 by A46;
            take y1,y2,y3,K;
            thus y1 = Py(x,z+1) & K > 2*z*y1 & y2 = Py(K,z+1) &
            y3 = Py(K*x,z+1) by A46;
            x is non trivial by A46,NEWTON03:def 1;
            then y1 >0 & (2*z) >0 by A46,XREAL_1:129,HILB10_1:13;
            then 2*z*y1 >0 by XREAL_1:129;
            then 2*z*y1 >=1 by NAT_1:14;
            then K > 1 by A46,XXREAL_0:2;
            then K is non trivial by NEWTON03:def 1;
            then
            A47: y2 >0 by A46,HILB10_1:13;
            per cases by A46;
            suppose (y*y2 >= y3)  & (2*y*y2 < 1*y2+2*y3 );
              then ( y*y2/y2 >= y3/y2) & ( 2*y*y2/y2 < (1*y2 + 2*y3)/y2 )
              by A47, XREAL_1:74,72;
              then ( y >= y3/y2) & ( 2*y < (1*y2 + 2*y3)/y2 )
              by XCMPLX_1:89, A47;
              then A48: ( y-y3/y2>= y3/y2 - y3/y2)&(2*y*1 < 1*y2/y2 + 2*y3/y2)
              by XREAL_1:9,XCMPLX_1:62;
              then ( y-y3/y2>= 0) & (2*y < 1 + 2*y3/y2) by XCMPLX_1:89,A47;
              then (2*y/2 < (1 + 2*y3/y2)/2) by XREAL_1:74;
              then (y/(2/2) < 1/2 + 2*y3/y2/2 );
              then A49: y < 1/2 + 2* y3/ (y2 * 2) by XCMPLX_1:78;
              2* y3/ (y2 * 2) = y3/y2 by XCMPLX_1:91;
              then (y - y3/y2 < 1/2 + y3/y2 - y3/y2 ) by A49,XREAL_1:9;
              hence thesis by A48;
            end;
            suppose A50: (y3 >= y*y2) & ( (-2)*y*y2 < 1*y2 + (-2)*y3 );
              then (y3/y2 >= y*y2/y2) & ( (-2)*y*y2 < 1*y2 + (-2)*y3 )
              by XREAL_1:72;
              then (y3/y2 >= y) by A47, XCMPLX_1:89;
              then A51: ( y3/y2 - y >= y - y) by XREAL_1:9;
              ( (-2)*y*y2 /y2 < ( 1*y2 + (-2)*y3) /y2 ) by A47,A50, XREAL_1:74;
              then ( (-2)*y < ( 1*y2 + (-2)*y3) /y2 ) by A47, XCMPLX_1:89;
              then ( (-2)*y < 1*y2/y2 + (-2)*y3 /y2 ) by XCMPLX_1:62;
              then ( (-2)*y < 1 + (-2)*y3 /y2 ) by A47, XCMPLX_1:89;
              then A52: ( (1 + (-2)*y3 /y2)/ (-2) < (-2)*y/(-2))
              by XREAL_1:75;
              A53: (-2)*y3/y2/(-2) = (-2)*y3/(y2*(-2)) by XCMPLX_1:78
              .= y3/y2 by XCMPLX_1:91;
              -1/2 + y3/y2 < y by A52,A53;
              then 1/2 > -(y-y3/y2) by XREAL_1:25,20;
              hence thesis by A51;
            end;
          end;
          hence thesis;
        end;
      end;
      then pn.i2 =(pn.i1) |^ (pn.i3) by HILB10_1:39;
      hence thesis by A43;
    end;
    WW c= DDR
    proof
      let o be object such that A54: o in WW;
      consider p such that
      A55: o=p & p.i2 =(p.i1)|^ (p.i3) by A54;
      set x=p.i1,y=p.i2,z=p.i3;
      per cases by A55,HILB10_1:39;
      suppose A56: (y = 1 & z = 0) or
        (x = 0 & y = 0 & z > 0) or
        (x = 1 & y = 1 & z > 0);
        reconsider Z=0,z1=z+1 as Element of NAT;
        set Y=<%Z%>^<%Z%>^<%Z%>^<%Z%>^<%z1%>^<%Z%>^<%Z%>;
        set PY = p^Y;
        A57: len p = n & len Y = 7 by CARD_1:def 7;
        A58: PY|n =p by A57,AFINSQ_1:57;
        P45[PY] or P036[PY] or P146[PY] by A56, A58,A1,Th4;
        then PY in DD;
        hence thesis by A55,A58;
      end;
      suppose A59: x > 1 & z > 0 & ex y1,y2,y3,K be Nat st
        y1 = Py(x,z+1) & K > 2*z*y1 & y2 = Py(K,z+1) & y3 = Py(K*x,z+1) &
        (0 <= y-y3/y2 <1/2 or 0 <= y3/y2 -y < 1/2);
        then consider y1,y2,y3,K be Nat such that
        A60: y1 = Py(x,z+1) & K > 2*z*y1 & y2 = Py(K,z+1) &
        y3 = Py(K*x,z+1) &
        (0 <= y-y3/y2 <1/2 or 0 <= y3/y2 -y < 1/2);
        reconsider y1,y2,y3,K,z1=z+1 as Element of NAT by ORDINAL1:def 12;
        reconsider Kx = K*x,yy2=y*y2 as Element of NAT;
        set Y=<%y1%>^<%y2%>^<%y3%>^<%K%>^<%z1%>^<%Kx%>^<%yy2%>;
        set PY = p^Y;
        A61: len p = n & len Y = 7 by CARD_1:def 7;
        A62: PY|n =p by A61,AFINSQ_1:57;
        4 in dom Y by A61,AFINSQ_1:66;
        then A63: PY.(n+4) =Y.4 by A61,AFINSQ_1:def 3
        .= z1 by AFINSQ_1:48;
        0 in dom Y by A61,AFINSQ_1:66;
        then A64: PY.(n+0) = Y.0 by A61,AFINSQ_1:def 3
        .= y1 by AFINSQ_1:48;
        3 in dom Y by A61,AFINSQ_1:66;
        then A65: PY.(n+3) = Y.3 by A61,AFINSQ_1:def 3
        .= K by AFINSQ_1:48;
        5 in dom Y by A61,AFINSQ_1:66;
        then A66:PY.(n+5) = Y.5 by A61,AFINSQ_1:def 3
        .= Kx by AFINSQ_1:48;
        x is non trivial by A59,NEWTON03:def 1;
        then y1 >0 & z >0 by A59,A60, HILB10_1:13;
        then z*y1 >0 by XREAL_1:129;
        then  2*(z*y1)>= 2*1 by XREAL_1:64,NAT_1:14;
        then A67: K >=1+1 by XXREAL_0:2,A60;
        then A68: K > 1 by XXREAL_0:2;
        1 in dom Y by A61,AFINSQ_1:66;
        then A69: PY.(n+1) = Y.1 by A61,AFINSQ_1:def 3
        .= y2 by AFINSQ_1:48;
        A70: K*x > 1 * 1 by A68, A59,XREAL_1:97;
        2 in dom Y by A61,AFINSQ_1:66;
        then A71: PY.(n+2) = Y.2 by A61,AFINSQ_1:def 3
        .= y3 by AFINSQ_1:48;
        6 in dom Y by A61,AFINSQ_1:66;
        then A72: PY.(n+6) = Y.6 by A61,AFINSQ_1:def 3
        .= yy2 by AFINSQ_1:48;
        x is non trivial by A59,NEWTON03:def 1;
        then y1 >0 & (2*z) >0 by A60,A59,XREAL_1:129,HILB10_1:13;
        then 2*z*y1 >0 by XREAL_1:129;
        then 2*z*y1 >=1 by NAT_1:14;
        then K > 1 by A60,XXREAL_0:2;
        then
        K is non trivial by NEWTON03:def 1;
        then A73: y2 >0 by A60,HILB10_1:13;
        PHIJK[PY]
        proof
          per cases by A60;
          suppose A74: ( y-y3/y2>= 0) & (y- y3/y2 < 1/2 );
            (y-y3/y2)*y2 >= 0 * y2 by A74;
            then y*y2 - y3/y2*y2 >= 0;
            then y*y2 - y3/(y2/y2) >= 0 by XCMPLX_1:82;
            then y*y2 - y3/1 >=0 by XCMPLX_1:60,A73;
            then A75: y*y2 - y3 + y3 >= 0 + y3 by XREAL_1:6;
            (y- y3/y2) *y2 < 1/2 *y2 by A74,XREAL_1:68,A73;
            then y*y2 -y3/y2 *y2 < 1/2 * y2;
            then y*y2 - y3/(y2/y2) < 1/2 * y2 by XCMPLX_1:82;
            then y*y2 - y3/1 < 1/2 * y2 by A73, XCMPLX_1:60;
            then (y*y2 - y3)*2 < 1/2 * y2 *2 by XREAL_1:68;
            then 2*y*y2 - 2*y3 + 2*y3 < 1*y2 + 2*y3 by XREAL_1:6;
            hence thesis by A72,A71,A69,A75;
          end;
          suppose A76: ( y3/y2 - y >= 0) & ( y3/y2 - y  < 1/2 );
            then y3/y2 - y + y >= 0 + y by XREAL_1:6;
            then y3/y2 * y2 >= y*y2 by XREAL_1:64;
            then y3/(y2/y2) >= y*y2 by XCMPLX_1:82;
            then A77:y3/1 >= y*y2 by A73, XCMPLX_1:60;
            (y3/y2 - y) *y2 < 1/2 *y2 by A76,XREAL_1:68,A73;
            then y3/y2*y2 - y*y2 < 1/2 * y2;
            then y3/(y2/y2) - y*y2 < 1/2 * y2 by XCMPLX_1:82;
            then y3/1 - y*y2 < 1/2 * y2 by A73, XCMPLX_1:60;
            then (y3 - y*y2 )*2 < 1/2 * y2 *2 by XREAL_1:68;
            then -2*y*y2 + 2*y3 - 2*y3 < 1 * y2 - 2*y3 by XREAL_1:14;
            hence thesis by A72,A71,A69,A77;
          end;
        end;
        then P26CDEFHIJKABG[PY] by A59,A63,A67,XXREAL_0:2,A65,A71,
          A60,A70,A66,A64,A62,A1,Th4,A69,A72;
        then PY in DD;
        hence thesis by A55,A62;
      end;
    end;
    hence thesis by A40,A41,XBOOLE_0:def 10;
  end;
end;
