reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i for Integer;
reserve r for Real;
reserve p for Prime;

theorem Th49:
  for n being composite Nat holds
  n <= 1105 & n divides 2|^n-2 implies n in {341,561,645,1105}
  proof
    let n be composite Nat;
    assume that
A1: n <= 1105 and
A2: n divides 2|^n-2;
A3: 2|^n,2 are_congruent_mod n by A2;
A4: 4 <= n by NUMBER02:40;
    assume
A5: not thesis;
LemF: for a st a <> 2 & a < n holds not 2|^n,a are_congruent_mod n
    proof
      2+1 <= n by A4,XXREAL_0:2;
      then 2 < n by NAT_1:13;
      then (2|^n) mod n = 2 by A3,NAT_6:12;
      hence thesis by NAT_6:12;
    end;
    per cases by A1;
    suppose n <= 104;
      then n = 4 or ... or n = 104 by A4;
      then per cases;
      suppose A: n = 4;
        2|^4 = 4*4+0 by Lm4;
        then 2|^4 mod 4 = 0;
        hence thesis by A3,A,NAT_6:12;
      end;
      suppose n = 5;
        hence thesis by XPRIMES1:5;
      end;
      suppose A: n = 6;
        2|^6 = 6*10+4 by Lm6;
        then 2|^6 mod 6 = 4 by NAT_D:def 2;
        hence thesis by A3,A,NAT_6:12;
      end;
      suppose n = 7;
        hence thesis by XPRIMES1:7;
      end;
      suppose A: n = 8;
        2|^8 = 8*32+0 by Lm8;
        then 2|^8 mod 8 = 0;
        hence thesis by A3,A,NAT_6:12;
      end;
      suppose A: n = 9;
        2|^9 = 9*56+8 by Lm9;
        then 2|^9 mod 9 = 8 by NAT_D:def 2;
        hence thesis by A3,A,NAT_6:12;
      end;
      suppose A: n = 10;
        2|^10 = 10*102+4 by Lm10;
        then 2|^10 mod 10 = 4 by NAT_D:def 2;
        hence thesis by A3,A,NAT_6:12;
      end;
      suppose n = 11;
        hence thesis by XPRIMES1:11;
      end;
      suppose A: n = 12;
        2|^12 = 12*341+4 by Lm12;
        then 2|^12 mod 12 = 4 by NAT_D:def 2;
        hence thesis by A3,A,NAT_6:12;
      end;
      suppose n = 13;
        hence thesis by XPRIMES1:13;
      end;
      suppose A: n = 14;
        2|^14 = 14*1170+4 by Lm14;
        then 2|^14 mod 14 = 4 by NAT_D:def 2;
        hence thesis by A3,A,NAT_6:12;
      end;
      suppose A: n = 15;
        2|^15 = 15*2184+8 by Lm15;
        then 2|^15 mod 15 = 8 by NAT_D:def 2;
        hence thesis by A3,A,NAT_6:12;
      end;
      suppose A: n = 16;
        2|^16 = 16*4096+0 by Lm16;
        then 2|^16 mod 16 = 0;
        hence thesis by A3,A,NAT_6:12;
      end;
      suppose n = 17;
        hence thesis by XPRIMES1:17;
      end;
      suppose A: n = 18;
        2|^4 = 18*0+16 & 16*16*16*16*2|^2 = 18*14563+10 by Lm4,Lm2;
        then 2|^(4*4+2),10 are_congruent_mod 18 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 19;
        hence thesis by XPRIMES1:19;
      end;
      suppose A: n = 20;
        2|^5 = 20*1+12 & 12*12*12*12*2|^0 = 20*1036+16 by Lm5,Lm1;
        then 2|^(5*4+0),16 are_congruent_mod 20 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 21;
        2|^5 = 21*1+11 & 11*11*11*11*2|^1 = 21*1394+8 by Lm5;
        then 2|^(5*4+1),8 are_congruent_mod 21 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 22;
        2|^5 = 22*1+10 & 10*10*10*10*2|^2 = 22*1818+4 by Lm5,Lm2;
        then 2|^(5*4+2),4 are_congruent_mod 22 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 23;
        hence thesis by XPRIMES1:23;
      end;
      suppose A: n = 24;
        2|^6 = 24*2+16 & 16*16*16*16*2|^0 = 24*2730+16 by Lm6,Lm1;
        then 2|^(6*4+0),16 are_congruent_mod 24 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 25;
        2|^6 = 25*2+14 & 14*14*14*14*2|^1 = 25*3073+7 by Lm6;
        then 2|^(6*4+1),7 are_congruent_mod 25 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 26;
        2|^6 = 26*2+12 & 12*12*12*12*2|^2 = 26*3190+4 by Lm6,Lm2;
        then 2|^(6*4+2),4 are_congruent_mod 26 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 27;
        2|^6 = 27*2+10 & 10*10*10*10*2|^3 = 27*2962+26 by Lm6,Lm3;
        then 2|^(6*4+3),26 are_congruent_mod 27 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 28;
        2|^7 = 28*4+16 & 16*16*16*16*2|^0 = 28*2340+16 by Lm7,Lm1;
        then 2|^(7*4+0),16 are_congruent_mod 28 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 29;
        hence thesis by XPRIMES1:29;
      end;
      suppose A: n = 30;
        2|^7 = 30*4+8 & 8*8*8*8*2|^2 = 30*546+4 by Lm7,Lm2;
        then 2|^(7*4+2),4 are_congruent_mod 30 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 31;
        hence thesis by XPRIMES1:31;
      end;
      suppose A: n = 32;
        2|^8 = 32*8+0 & 0*0*0*0*2|^0 = 32*0+0 by Lm8;
        then 2|^(8*4+0),0 are_congruent_mod 32 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 33;
        2|^8 = 33*7+25 & 25*25*25*25*2|^1 = 33*23674+8 by Lm8;
        then 2|^(8*4+1),8 are_congruent_mod 33 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 34;
        2|^8 = 34*7+18 & 18*18*18*18*2|^2 = 34*12350+4 by Lm8,Lm2;
        then 2|^(8*4+2),4 are_congruent_mod 34 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 35;
        2|^8 = 35*7+11 & 11*11*11*11*2|^3 = 35*3346+18 by Lm8,Lm3;
        then 2|^(8*4+3),18 are_congruent_mod 35 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 36;
        2|^9 = 36*14+8 & 8*8*8*8*2|^0 = 36*113+28 by Lm9,Lm1;
        then 2|^(9*4+0),28 are_congruent_mod 36 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 37;
        hence thesis by XPRIMES1:37;
      end;
      suppose A: n = 38;
        2|^9 = 38*13+18 & 18*18*18*18*2|^2 = 38*11050+4 by Lm9,Lm2;
        then 2|^(9*4+2),4 are_congruent_mod 38 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 39;
        2|^9 = 39*13+5 & 5*5*5*5*2|^3 = 39*128+8 by Lm9,Lm3;
        then 2|^(9*4+3),8 are_congruent_mod 39 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 40;
        2|^10 = 40*25+24 & 24*24*24*24*2|^0 = 40*8294+16 by Lm10,Lm1;
        then 2|^(10*4+0),16 are_congruent_mod 40 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 41;
        hence thesis by XPRIMES1:41;
      end;
      suppose A: n = 42;
        2|^10 = 42*24+16 & 16*16*16*16*2|^2 = 42*6241+22 by Lm10,Lm2;
        then 2|^(10*4+2),22 are_congruent_mod 42 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 43;
        hence thesis by XPRIMES1:43;
      end;
      suppose A: n = 44;
        2|^11 = 44*46+24 & 24*24*24*24*2|^0 = 44*7540+16 by Lm11,Lm1;
        then 2|^(11*4+0),16 are_congruent_mod 44 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 45;
        2|^11 = 45*45+23 & 23*23*23*23*2|^1 = 45*12437+17 by Lm11;
        then 2|^(11*4+1),17 are_congruent_mod 45 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 46;
        2|^11 = 46*44+24 & 24*24*24*24*2|^2 = 46*28850+4 by Lm11,Lm2;
        then 2|^(11*4+2),4 are_congruent_mod 46 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 47;
        hence thesis by XPRIMES1:47;
      end;
      suppose A: n = 48;
        2|^12 = 48*85+16 & 16*16*16*16*2|^0 = 48*1365+16 by Lm12,Lm1;
        then 2|^(12*4+0),16 are_congruent_mod 48 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 49;
        2|^12 = 49*83+29 & 29*29*29*29*2|^1 = 49*28868+30 by Lm12;
        then 2|^(12*4+1),30 are_congruent_mod 49 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 50;
        2|^12 = 50*81+46 & 46*46*46*46*2|^2 = 50*358196+24 by Lm12,Lm2;
        then 2|^(12*4+2),24 are_congruent_mod 50 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 51;
        2|^12 = 51*80+16 & 16*16*16*16*2|^3 = 51*10280+8 by Lm12,Lm3;
        then 2|^(12*4+3),8 are_congruent_mod 51 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 52;
        2|^13 = 52*157+28 & 28*28*28*28*2|^0 = 52*11820+16 by Lm13,Lm1;
        then 2|^(13*4+0),16 are_congruent_mod 52 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 53;
        hence thesis by XPRIMES1:53;
      end;
      suppose A: n = 54;
        2|^13 = 54*151+38 & 38*38*38*38*2|^2 = 54*154454+28 by Lm13,Lm2;
        then 2|^(13*4+2),28 are_congruent_mod 54 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 55;
        2|^13 = 55*148+52 & 52*52*52*52*2|^3 = 55*1063507+43 by Lm13,Lm3;
        then 2|^(13*4+3),43 are_congruent_mod 55 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 56;
        2|^14 = 56*292+32 & 32*32*32*32*2|^0 = 56*18724+32 by Lm14,Lm1;
        then 2|^(14*4+0),32 are_congruent_mod 56 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 57;
        2|^14 = 57*287+25 & 25*25*25*25*2|^1 = 57*13706+8 by Lm14;
        then 2|^(14*4+1),8 are_congruent_mod 57 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 58;
        2|^14 = 58*282+28 & 28*28*28*28*2|^2 = 58*42390+4 by Lm14,Lm2;
        then 2|^(14*4+2),4 are_congruent_mod 58 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 59;
        hence thesis by XPRIMES1:59;
      end;
      suppose A: n = 60;
        2|^15 = 60*546+8 & 8*8*8*8*2|^0 = 60*68+16 by Lm15,Lm1;
        then 2|^(15*4+0),16 are_congruent_mod 60 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 61;
        hence thesis by XPRIMES1:61;
      end;
      suppose A: n = 62;
        2|^15 = 62*528+32 & 32*32*32*32*2|^2 = 62*67650+4 by Lm15,Lm2;
        then 2|^(15*4+2),4 are_congruent_mod 62 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 63;
        2|^15 = 63*520+8 & 8*8*8*8*2|^3 = 63*520+8 by Lm15,Lm3;
        then 2|^(15*4+3),8 are_congruent_mod 63 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 64;
        2|^16 = 64*1024+0 & 0*0*0*0*2|^0 = 64*0+0 by Lm16;
        then 2|^(16*4+0),0 are_congruent_mod 64 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 65;
        2|^16 = 65*1008+16 & 16*16*16*16*2|^1 = 65*2016+32 by Lm16;
        then 2|^(16*4+1),32 are_congruent_mod 65 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose A: n = 66;
        2|^16 = 66*992+64 & 64*64*64*64*2|^2 = 66*1016800+64 by Lm16,Lm2;
        then 2|^(16*4+2),64 are_congruent_mod 66 by Lem1;
        hence thesis by A,LemF;
      end;
      suppose n = 67;
        hence thesis by XPRIMES1:67;
      end;
      suppose A: n = 68;
        2|^4 = 68*0+16 & 16*16*16*16*2|^1 = 68*1927+36 by Lm4;
        then 2|^(4*4+1),36 are_congruent_mod 68 &
          36*36*36*36*2|^0 = 68*24700+16 by Lem1,Lm1;
        then 2|^(17*4+0),16 are_congruent_mod 68 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 69;
        2|^4 = 69*0+16 & 16*16*16*16*2|^1 = 69*1899+41 by Lm4;
        then 2|^(4*4+1),41 are_congruent_mod 69 &
          41*41*41*41*2|^1 = 69*81906+8 by Lem1;
        then 2|^(17*4+1),8 are_congruent_mod 69 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 70;
        2|^4 = 70*0+16 & 16*16*16*16*2|^1 = 70*1872+32 by Lm4;
        then 2|^(4*4+1),32 are_congruent_mod 70 &
          32*32*32*32*2|^2 = 70*59918+44 by Lem1,Lm2;
        then 2|^(17*4+2),44 are_congruent_mod 70 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 71;
        hence thesis by XPRIMES1:71;
      end;
      suppose A: n = 72;
        2|^4 = 72*0+16 & 16*16*16*16*2|^2 = 72*3640+64 by Lm4,Lm2;
        then 2|^(4*4+2),64 are_congruent_mod 72 &
          64*64*64*64*2|^0 = 72*233016+64 by Lem1,Lm1;
        then 2|^(18*4+0),64 are_congruent_mod 72 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 73;
        hence thesis by XPRIMES1:73;
      end;
      suppose A: n = 74;
        2|^4 = 74*0+16 & 16*16*16*16*2|^2 = 74*3542+36 by Lm4,Lm2;
        then 2|^(4*4+2),36 are_congruent_mod 74 &
          36*36*36*36*2|^2 = 74*90790+4 by Lem1;
        then 2|^(18*4+2),4 are_congruent_mod 74 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 75;
        2|^4 = 75*0+16 & 16*16*16*16*2|^2 = 75*3495+19 by Lm4,Lm2;
        then 2|^(4*4+2),19 are_congruent_mod 75 &
          19*19*19*19*2|^3 = 75*13900+68 by Lem1,Lm3;
        then 2|^(18*4+3),68 are_congruent_mod 75 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 76;
        2|^4 = 76*0+16 & 16*16*16*16*2|^3 = 76*6898+40 by Lm4,Lm3;
        then 2|^(4*4+3),40 are_congruent_mod 76 &
          40*40*40*40*2|^0 = 76*33684+16 by Lem1,Lm1;
        then 2|^(19*4+0),16 are_congruent_mod 76 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 77;
        2|^4 = 77*0+16 & 16*16*16*16*2|^3 = 77*6808+72 by Lm4,Lm3;
        then 2|^(4*4+3),72 are_congruent_mod 77 &
          72*72*72*72*2|^1 = 77*698022+18 by Lem1;
        then 2|^(19*4+1),18 are_congruent_mod 77 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 78;
        2|^4 = 78*0+16 & 16*16*16*16*2|^3 = 78*6721+50 by Lm4,Lm3;
        then 2|^(4*4+3),50 are_congruent_mod 78 &
          50*50*50*50*2|^2 = 78*320512+64 by Lem1,Lm2;
        then 2|^(19*4+2),64 are_congruent_mod 78 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 79;
        hence thesis by XPRIMES1:79;
      end;
      suppose A: n = 80;
        2|^5 = 80*0+32 & 32*32*32*32*2|^0 = 80*13107+16 by Lm5,Lm1;
        then 2|^(5*4+0),16 are_congruent_mod 80 &
          16*16*16*16*2|^0 = 80*819+16 by Lem1;
        then 2|^(20*4+0),16 are_congruent_mod 80 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 81;
        2|^5 = 81*0+32 & 32*32*32*32*2|^0 = 81*12945+31 by Lm5,Lm1;
        then 2|^(5*4+0),31 are_congruent_mod 81 &
          31*31*31*31*2|^1 = 81*22802+80 by Lem1;
        then 2|^(20*4+1),80 are_congruent_mod 81 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 82;
        2|^5 = 82*0+32 & 32*32*32*32*2|^0 = 82*12787+42 by Lm5,Lm1;
        then 2|^(5*4+0),42 are_congruent_mod 82 &
          42*42*42*42*2|^2 = 82*151790+4 by Lem1,Lm2;
        then 2|^(20*4+2),4 are_congruent_mod 82 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 83;
        hence thesis by XPRIMES1:83;
      end;
      suppose A: n = 84;
        2|^5 = 84*0+32 & 32*32*32*32*2|^1 = 84*24966+8 by Lm5;
        then 2|^(5*4+1),8 are_congruent_mod 84 &
          8*8*8*8*2|^0 = 84*48+64 by Lem1,Lm1;
        then 2|^(21*4+0),64 are_congruent_mod 84 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 85;
        2|^5 = 85*0+32 & 32*32*32*32*2|^1 = 85*24672+32 by Lm5;
        then 2|^(5*4+1),32 are_congruent_mod 85 &
          32*32*32*32*2|^1 = 85*24672+32 by Lem1;
        then 2|^(21*4+1),32 are_congruent_mod 85 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 86;
        2|^5 = 86*0+32 & 32*32*32*32*2|^1 = 86*24385+42 by Lm5;
        then 2|^(5*4+1),42 are_congruent_mod 86 &
          42*42*42*42*2|^2 = 86*144730+4 by Lem1,Lm2;
        then 2|^(21*4+2),4 are_congruent_mod 86 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 87;
        2|^5 = 87*0+32 & 32*32*32*32*2|^1 = 87*24105+17 by Lm5;
        then 2|^(5*4+1),17 are_congruent_mod 87 &
          17*17*17*17*2|^3 = 87*7680+8 by Lem1,Lm3;
        then 2|^(21*4+3),8 are_congruent_mod 87 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 88;
        2|^5 = 88*0+32 & 32*32*32*32*2|^2 = 88*47662+48 by Lm5,Lm2;
        then 2|^(5*4+2),48 are_congruent_mod 88 &
          48*48*48*48*2|^0 = 88*60322+80 by Lem1,Lm1;
        then 2|^(22*4+0),80 are_congruent_mod 88 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 89;
        hence thesis by XPRIMES1:89;
      end;
      suppose A: n = 90;
        2|^5 = 90*0+32 & 32*32*32*32*2|^2 = 90*46603+34 by Lm5,Lm2;
        then 2|^(5*4+2),34 are_congruent_mod 90 &
          34*34*34*34*2|^2 = 90*59392+64 by Lem1;
        then 2|^(22*4+2),64 are_congruent_mod 90 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 91;
        2|^5 = 91*0+32 & 32*32*32*32*2|^2 = 91*46091+23 by Lm5,Lm2;
        then 2|^(5*4+2),23 are_congruent_mod 91 &
          23*23*23*23*2|^3 = 91*24601+37 by Lem1,Lm3;
        then 2|^(22*4+3),37 are_congruent_mod 91 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 92;
        2|^5 = 92*0+32 & 32*32*32*32*2|^3 = 92*91180+48 by Lm5,Lm3;
        then 2|^(5*4+3),48 are_congruent_mod 92 &
          48*48*48*48*2|^0 = 92*57700+16 by Lem1,Lm1;
        then 2|^(23*4+0),16 are_congruent_mod 92 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 93;
        2|^5 = 93*0+32 & 32*32*32*32*2|^3 = 93*90200+8 by Lm5,Lm3;
        then 2|^(5*4+3),8 are_congruent_mod 93 &
          8*8*8*8*2|^1 = 93*88+8 by Lem1;
        then 2|^(23*4+1),8 are_congruent_mod 93 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 94;
        2|^5 = 94*0+32 & 32*32*32*32*2|^3 = 94*89240+48 by Lm5,Lm3;
        then 2|^(5*4+3),48 are_congruent_mod 94 &
          48*48*48*48*2|^2 = 94*225890+4 by Lem1,Lm2;
        then 2|^(23*4+2),4 are_congruent_mod 94 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 95;
        2|^5 = 95*0+32 & 32*32*32*32*2|^3 = 95*88301+13 by Lm5,Lm3;
        then 2|^(5*4+3),13 are_congruent_mod 95 &
          13*13*13*13*2|^3 = 95*2405+13 by Lem1;
        then 2|^(23*4+3),13 are_congruent_mod 95 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 96;
        2|^6 = 96*0+64 & 64*64*64*64*2|^0 = 96*174762+64 by Lm6,Lm1;
        then 2|^(6*4+0),64 are_congruent_mod 96 &
          64*64*64*64*2|^0 = 96*174762+64 by Lem1;
        then 2|^(24*4+0),64 are_congruent_mod 96 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 97;
        hence thesis by XPRIMES1:97;
      end;
      suppose A: n = 98;
        2|^6 = 98*0+64 & 64*64*64*64*2|^0 = 98*171196+8 by Lm6,Lm1;
        then 2|^(6*4+0),8 are_congruent_mod 98 &
          8*8*8*8*2|^2 = 98*167+18 by Lem1,Lm2;
        then 2|^(24*4+2),18 are_congruent_mod 98 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 99;
        2|^6 = 99*0+64 & 64*64*64*64*2|^0 = 99*169466+82 by Lm6,Lm1;
        then 2|^(6*4+0),82 are_congruent_mod 99 &
          82*82*82*82*2|^3 = 99*3653509+17 by Lem1,Lm3;
        then 2|^(24*4+3),17 are_congruent_mod 99 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 100;
        2|^6 = 100*0+64 & 64*64*64*64*2|^1 = 100*335544+32 by Lm6;
        then 2|^(6*4+1),32 are_congruent_mod 100 &
          32*32*32*32*2|^0 = 100*10485+76 by Lem1,Lm1;
        then 2|^(25*4+0),76 are_congruent_mod 100 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 101;
        hence thesis by XPRIMES1:101;
      end;
      suppose A: n = 102;
        2|^6 = 102*0+64 & 64*64*64*64*2|^1 = 102*328965+2 by Lm6;
        then 2|^(6*4+1),2 are_congruent_mod 102 &
          2*2*2*2*2|^2 = 102*0+64 by Lem1,Lm2;
        then 2|^(25*4+2),64 are_congruent_mod 102 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 103;
        hence thesis by XPRIMES1:103;
      end;
      suppose A: n = 104;
        2|^6 = 104*0+64 & 64*64*64*64*2|^2 = 104*645277+56 by Lm6,Lm2;
        then 2|^(6*4+2),56 are_congruent_mod 104 &
          56*56*56*56*2|^0 = 104*94562+48 by Lem1,Lm1;
        then 2|^(26*4+0),48 are_congruent_mod 104 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 104 < n & n <= 204;
      then 104+1 <= n & n <= 204 by NAT_1:13;
      then n = 105 or ... or n = 204;
      then per cases;
      suppose A: n = 105;
        2|^6 = 105*0+64 & 64*64*64*64*2|^2 = 105*639132+4 by Lm6,Lm2;
        then 2|^(6*4+2),4 are_congruent_mod 105 &
          4*4*4*4*2|^1 = 105*4+92 by Lem1;
        then 2|^(26*4+1),92 are_congruent_mod 105 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 106;
        2|^6 = 106*0+64 & 64*64*64*64*2|^2 = 106*633102+52 by Lm6,Lm2;
        then 2|^(6*4+2),52 are_congruent_mod 106 &
          52*52*52*52*2|^2 = 106*275910+4 by Lem1;
        then 2|^(26*4+2),4 are_congruent_mod 106 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 107;
        hence thesis by XPRIMES1:107;
      end;
      suppose A: n = 108;
        2|^6 = 108*0+64 & 64*64*64*64*2|^3 = 108*1242756+80 by Lm6,Lm3;
        then 2|^(6*4+3),80 are_congruent_mod 108 &
          80*80*80*80*2|^0 = 108*379259+28 by Lem1,Lm1;
        then 2|^(27*4+0),28 are_congruent_mod 108 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 109;
        hence thesis by XPRIMES1:109;
      end;
      suppose A: n = 110;
        2|^6 = 110*0+64 & 64*64*64*64*2|^3 = 110*1220161+18 by Lm6,Lm3;
        then 2|^(6*4+3),18 are_congruent_mod 110 &
          18*18*18*18*2|^2 = 110*3817+34 by Lem1,Lm2;
        then 2|^(27*4+2),34 are_congruent_mod 110 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 111;
        2|^6 = 111*0+64 & 64*64*64*64*2|^3 = 111*1209168+80 by Lm6,Lm3;
        then 2|^(6*4+3),80 are_congruent_mod 111 &
          80*80*80*80*2|^3 = 111*2952072+8 by Lem1;
        then 2|^(27*4+3),8 are_congruent_mod 111 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 112;
        2|^7 = 112*1+16 & 16*16*16*16*2|^0 = 112*585+16 by Lm7,Lm1;
        then 2|^(7*4+0),16 are_congruent_mod 112 &
          16*16*16*16*2|^0 = 112*585+16 by Lem1;
        then 2|^(28*4+0),16 are_congruent_mod 112 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 113;
        hence thesis by XPRIMES1:113;
      end;
      suppose A: n = 114;
        2|^7 = 114*1+14 & 14*14*14*14*2|^0 = 114*336+112 by Lm7,Lm1;
        then 2|^(7*4+0),112 are_congruent_mod 114 &
          112*112*112*112*2|^2 = 114*5521120+64 by Lem1,Lm2;
        then 2|^(28*4+2),64 are_congruent_mod 114 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 115;
        2|^7 = 115*1+13 & 13*13*13*13*2|^0 = 115*248+41 by Lm7,Lm1;
        then 2|^(7*4+0),41 are_congruent_mod 115 &
          41*41*41*41*2|^3 = 115*196574+78 by Lem1,Lm3;
        then 2|^(28*4+3),78 are_congruent_mod 115 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 116;
        2|^7 = 116*1+12 & 12*12*12*12*2|^1 = 116*357+60 by Lm7;
        then 2|^(7*4+1),60 are_congruent_mod 116 &
          60*60*60*60*2|^0 = 116*111724+16 by Lem1,Lm1;
        then 2|^(29*4+0),16 are_congruent_mod 116 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 117;
        2|^7 = 117*1+11 & 11*11*11*11*2|^1 = 117*250+32 by Lm7;
        then 2|^(7*4+1),32 are_congruent_mod 117 &
          32*32*32*32*2|^1 = 117*17924+44 by Lem1;
        then 2|^(29*4+1),44 are_congruent_mod 117 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 118;
        2|^7 = 118*1+10 & 10*10*10*10*2|^1 = 118*169+58 by Lm7;
        then 2|^(7*4+1),58 are_congruent_mod 118 &
          58*58*58*58*2|^2 = 118*383610+4 by Lem1,Lm2;
        then 2|^(29*4+2),4 are_congruent_mod 118 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 119;
        2|^7 = 119*1+9 & 9*9*9*9*2|^1 = 119*110+32 by Lm7;
        then 2|^(7*4+1),32 are_congruent_mod 119 &
          32*32*32*32*2|^3 = 119*70492+60 by Lem1,Lm3;
        then 2|^(29*4+3),60 are_congruent_mod 119 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 120;
        2|^7 = 120*1+8 & 8*8*8*8*2|^2 = 120*136+64 by Lm7,Lm2;
        then 2|^(7*4+2),64 are_congruent_mod 120 &
          64*64*64*64*2|^0 = 120*139810+16 by Lem1,Lm1;
        then 2|^(30*4+0),16 are_congruent_mod 120 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 121;
        2|^7 = 121*1+7 & 7*7*7*7*2|^2 = 121*79+45 by Lm7,Lm2;
        then 2|^(7*4+2),45 are_congruent_mod 121 &
          45*45*45*45*2|^1 = 121*67778+112 by Lem1;
        then 2|^(30*4+1),112 are_congruent_mod 121 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 122;
        2|^7 = 122*1+6 & 6*6*6*6*2|^2 = 122*42+60 by Lm7,Lm2;
        then 2|^(7*4+2),60 are_congruent_mod 122 &
          60*60*60*60*2|^2 = 122*424918+4 by Lem1;
        then 2|^(30*4+2),4 are_congruent_mod 122 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 123;
        2|^7 = 123*1+5 & 5*5*5*5*2|^2 = 123*20+40 by Lm7,Lm2;
        then 2|^(7*4+2),40 are_congruent_mod 123 &
          40*40*40*40*2|^3 = 123*166504+8 by Lem1,Lm3;
        then 2|^(30*4+3),8 are_congruent_mod 123 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 124;
        2|^7 = 124*1+4 & 4*4*4*4*2|^3 = 124*16+64 by Lm7,Lm3;
        then 2|^(7*4+3),64 are_congruent_mod 124 &
          64*64*64*64*2|^0 = 124*135300+16 by Lem1,Lm1;
        then 2|^(31*4+0),16 are_congruent_mod 124 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 125;
        2|^7 = 125*1+3 & 3*3*3*3*2|^3 = 125*5+23 by Lm7,Lm3;
        then 2|^(7*4+3),23 are_congruent_mod 125 &
          23*23*23*23*2|^1 = 125*4477+57 by Lem1;
        then 2|^(31*4+1),57 are_congruent_mod 125 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 126;
        2|^7 = 126*1+2 & 2*2*2*2*2|^3 = 126*1+2 by Lm7,Lm3;
        then 2|^(7*4+3),2 are_congruent_mod 126 &
          2*2*2*2*2|^2 = 126*0+64 by Lem1,Lm2;
        then 2|^(31*4+2),64 are_congruent_mod 126 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 127;
        hence thesis by XPRIMES1:127;
      end;
      suppose A: n = 128;
        2|^8 = 128*2+0 & 0*0*0*0*2|^0 = 128*0+0 by Lm8;
        then 2|^(8*4+0),0 are_congruent_mod 128 &
          0*0*0*0*2|^0 = 128*0+0 by Lem1;
        then 2|^(32*4+0),0 are_congruent_mod 128 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 129;
        2|^8 = 129*1+127 & 127*127*127*127*2|^0 = 129*2016625+16 by Lm8,Lm1;
        then 2|^(8*4+0),16 are_congruent_mod 129 &
          16*16*16*16*2|^1 = 129*1016+8 by Lem1;
        then 2|^(32*4+1),8 are_congruent_mod 129 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 130;
        2|^8 = 130*1+126 & 126*126*126*126*2|^0 = 130*1938825+126 by Lm8,Lm1;
        then 2|^(8*4+0),126 are_congruent_mod 130 &
          126*126*126*126*2|^2 = 130*7755303+114 by Lem1,Lm2;
        then 2|^(32*4+2),114 are_congruent_mod 130 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 131;
        hence thesis by XPRIMES1:131;
      end;
      suppose A: n = 132;
        2|^8 = 132*1+124 & 124*124*124*124*2|^1 = 132*3582142+8 by Lm8;
        then 2|^(8*4+1),8 are_congruent_mod 132 &
          8*8*8*8*2|^0 = 132*31+4 by Lem1,Lm1;
        then 2|^(33*4+0),4 are_congruent_mod 132 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 133;
        2|^8 = 133*1+123 & 123*123*123*123*2|^1 = 133*3441904+50 by Lm8;
        then 2|^(8*4+1),50 are_congruent_mod 133 &
          50*50*50*50*2|^1 = 133*93984+128 by Lem1;
        then 2|^(33*4+1),128 are_congruent_mod 133 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 134;
        2|^8 = 134*1+122 & 122*122*122*122*2|^1 = 134*3306469+66 by Lm8;
        then 2|^(8*4+1),66 are_congruent_mod 134 &
          66*66*66*66*2|^2 = 134*566410+4 by Lem1,Lm2;
        then 2|^(33*4+2),4 are_congruent_mod 134 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 135;
        2|^8 = 135*1+121 & 121*121*121*121*2|^1 = 135*3175687+17 by Lm8;
        then 2|^(8*4+1),17 are_congruent_mod 135 &
          17*17*17*17*2|^3 = 135*4949+53 by Lem1,Lm3;
        then 2|^(33*4+3),53 are_congruent_mod 135 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 136;
        2|^8 = 136*1+120 & 120*120*120*120*2|^2 = 136*6098823+72 by Lm8,Lm2;
        then 2|^(8*4+2),72 are_congruent_mod 136 &
          72*72*72*72*2|^0 = 136*197601+120 by Lem1,Lm1;
        then 2|^(34*4+0),120 are_congruent_mod 136 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 137;
        hence thesis by XPRIMES1:137;
      end;
      suppose A: n = 138;
        2|^8 = 138*1+118 & 118*118*118*118*2|^2 = 138*5619645+94 by Lm8,Lm2;
        then 2|^(8*4+2),94 are_congruent_mod 138 &
          94*94*94*94*2|^2 = 138*2263040+64 by Lem1;
        then 2|^(34*4+2),64 are_congruent_mod 138 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 139;
        hence thesis by XPRIMES1:139;
      end;
      suppose A: n = 140;
        2|^8 = 140*1+116 & 116*116*116*116*2|^3 = 140*10346510+88 by Lm8,Lm3;
        then 2|^(8*4+3),88 are_congruent_mod 140 &
          88*88*88*88*2|^0 = 140*428353+116 by Lem1,Lm1;
        then 2|^(35*4+0),116 are_congruent_mod 140 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 141;
        2|^8 = 141*1+115 & 115*115*115*115*2|^3 = 141*9923439+101 by Lm8,Lm3;
        then 2|^(8*4+3),101 are_congruent_mod 141 &
          101*101*101*101*2|^1 = 141*1476034+8 by Lem1;
        then 2|^(35*4+1),8 are_congruent_mod 141 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 142;
        2|^8 = 142*1+114 & 114*114*114*114*2|^3 = 142*9515268+72 by Lm8,Lm3;
        then 2|^(8*4+3),72 are_congruent_mod 142 &
          72*72*72*72*2|^2 = 142*757010+4 by Lem1,Lm2;
        then 2|^(35*4+2),4 are_congruent_mod 142 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 143;
        2|^8 = 143*1+113 & 113*113*113*113*2|^3 = 143*9121530+98 by Lm8,Lm3;
        then 2|^(8*4+3),98 are_congruent_mod 143 &
          98*98*98*98*2|^3 = 143*5160101+85 by Lem1;
        then 2|^(35*4+3),85 are_congruent_mod 143 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 144;
        2|^9 = 144*3+80 & 80*80*80*80*2|^0 = 144*284444+64 by Lm9,Lm1;
        then 2|^(9*4+0),64 are_congruent_mod 144 &
          64*64*64*64*2|^0 = 144*116508+64 by Lem1;
        then 2|^(36*4+0),64 are_congruent_mod 144 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 145;
        2|^9 = 145*3+77 & 77*77*77*77*2|^0 = 145*242434+111 by Lm9,Lm1;
        then 2|^(9*4+0),111 are_congruent_mod 145 &
          111*111*111*111*2|^1 = 145*2093890+32 by Lem1;
        then 2|^(36*4+1),32 are_congruent_mod 145 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 146;
        2|^9 = 146*3+74 & 74*74*74*74*2|^0 = 146*205387+74 by Lm9,Lm1;
        then 2|^(9*4+0),74 are_congruent_mod 146 &
          74*74*74*74*2|^2 = 146*821550+4 by Lem1,Lm2;
        then 2|^(36*4+2),4 are_congruent_mod 146 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 147;
        2|^9 = 147*3+71 & 71*71*71*71*2|^0 = 147*172868+85 by Lm9,Lm1;
        then 2|^(9*4+0),85 are_congruent_mod 147 &
          85*85*85*85*2|^3 = 147*2840850+50 by Lem1,Lm3;
        then 2|^(36*4+3),50 are_congruent_mod 147 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 148;
        2|^9 = 148*3+68 & 68*68*68*68*2|^1 = 148*288937+76 by Lm9;
        then 2|^(9*4+1),76 are_congruent_mod 148 &
          76*76*76*76*2|^0 = 148*225420+16 by Lem1,Lm1;
        then 2|^(37*4+0),16 are_congruent_mod 148 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 149;
        hence thesis by XPRIMES1:149;
      end;
      suppose A: n = 150;
        2|^9 = 150*3+62 & 62*62*62*62*2|^1 = 150*197017+122 by Lm9;
        then 2|^(9*4+1),122 are_congruent_mod 150 &
          122*122*122*122*2|^2 = 150*5907558+124 by Lem1,Lm2;
        then 2|^(37*4+2),124 are_congruent_mod 150 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 151;
        hence thesis by XPRIMES1:151;
      end;
      suppose A: n = 152;
        2|^9 = 152*3+56 & 56*56*56*56*2|^2 = 152*258802+80 by Lm9,Lm2;
        then 2|^(9*4+2),80 are_congruent_mod 152 &
          80*80*80*80*2|^0 = 152*269473+104 by Lem1,Lm1;
        then 2|^(38*4+0),104 are_congruent_mod 152 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 153;
        2|^9 = 153*3+53 & 53*53*53*53*2|^2 = 153*206287+13 by Lm9,Lm2;
        then 2|^(9*4+2),13 are_congruent_mod 153 &
          13*13*13*13*2|^1 = 153*373+53 by Lem1;
        then 2|^(38*4+1),53 are_congruent_mod 153 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 154;
        2|^9 = 154*3+50 & 50*50*50*50*2|^2 = 154*162337+102 by Lm9,Lm2;
        then 2|^(9*4+2),102 are_congruent_mod 154 &
          102*102*102*102*2|^2 = 154*2811512+16 by Lem1;
        then 2|^(38*4+2),16 are_congruent_mod 154 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 155;
        2|^9 = 155*3+47 & 47*47*47*47*2|^2 = 155*125927+39 by Lm9,Lm2;
        then 2|^(9*4+2),39 are_congruent_mod 155 &
          39*39*39*39*2|^3 = 155*119403+63 by Lem1,Lm3;
        then 2|^(38*4+3),63 are_congruent_mod 155 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 156;
        2|^9 = 156*3+44 & 44*44*44*44*2|^3 = 156*192210+8 by Lm9,Lm3;
        then 2|^(9*4+3),8 are_congruent_mod 156 &
          8*8*8*8*2|^0 = 156*26+40 by Lem1,Lm1;
        then 2|^(39*4+0),40 are_congruent_mod 156 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 157;
        hence thesis by XPRIMES1:157;
      end;
      suppose A: n = 158;
        2|^9 = 158*3+38 & 38*38*38*38*2|^3 = 158*105576+80 by Lm9,Lm3;
        then 2|^(9*4+3),80 are_congruent_mod 158 &
          80*80*80*80*2|^2 = 158*1036962+4 by Lem1,Lm2;
        then 2|^(39*4+2),4 are_congruent_mod 158 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 159;
        2|^9 = 159*3+35 & 35*35*35*35*2|^3 = 159*75503+23 by Lm9,Lm3;
        then 2|^(9*4+3),23 are_congruent_mod 159 &
          23*23*23*23*2|^3 = 159*14080+8 by Lem1;
        then 2|^(39*4+3),8 are_congruent_mod 159 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 160;
        2|^10 = 160*6+64 & 64*64*64*64*2|^0 = 160*104857+96 by Lm10,Lm1;
        then 2|^(10*4+0),96 are_congruent_mod 160 &
          96*96*96*96*2|^0 = 160*530841+96 by Lem1;
        then 2|^(40*4+0),96 are_congruent_mod 160 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 161;
        2|^10 = 161*6+58 & 58*58*58*58*2|^0 = 161*70288+128 by Lm10,Lm1;
        then 2|^(10*4+0),128 are_congruent_mod 161 &
          128*128*128*128*2|^1 = 161*3334601+151 by Lem1;
        then 2|^(40*4+1),151 are_congruent_mod 161 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 162;
        2|^10 = 162*6+52 & 52*52*52*52*2|^0 = 162*45133+70 by Lm10,Lm1;
        then 2|^(10*4+0),70 are_congruent_mod 162 &
          70*70*70*70*2|^2 = 162*592839+82 by Lem1,Lm2;
        then 2|^(40*4+2),82 are_congruent_mod 162 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 163;
        hence thesis by XPRIMES1:163;
      end;
      suppose A: n = 164;
        2|^10 = 164*6+40 & 40*40*40*40*2|^1 = 164*31219+84 by Lm10;
        then 2|^(10*4+1),84 are_congruent_mod 164 &
          84*84*84*84*2|^0 = 164*303580+16 by Lem1,Lm1;
        then 2|^(41*4+0),16 are_congruent_mod 164 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 165;
        2|^10 = 165*6+34 & 34*34*34*34*2|^1 = 165*16198+2 by Lm10;
        then 2|^(10*4+1),2 are_congruent_mod 165 &
          2*2*2*2*2|^1 = 165*0+32 by Lem1;
        then 2|^(41*4+1),32 are_congruent_mod 165 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 166;
        2|^10 = 166*6+28 & 28*28*28*28*2|^1 = 166*7405+82 by Lm10;
        then 2|^(10*4+1),82 are_congruent_mod 166 &
          82*82*82*82*2|^2 = 166*1089450+4 by Lem1,Lm2;
        then 2|^(41*4+2),4 are_congruent_mod 166 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 167;
        hence thesis by XPRIMES1:167;
      end;
      suppose A: n = 168;
        2|^10 = 168*6+16 & 16*16*16*16*2|^2 = 168*1560+64 by Lm10,Lm2;
        then 2|^(10*4+2),64 are_congruent_mod 168 &
          64*64*64*64*2|^0 = 168*99864+64 by Lem1,Lm1;
        then 2|^(42*4+0),64 are_congruent_mod 168 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 169;
        2|^10 = 169*6+10 & 10*10*10*10*2|^2 = 169*236+116 by Lm10,Lm2;
        then 2|^(10*4+2),116 are_congruent_mod 169 &
          116*116*116*116*2|^1 = 169*2142768+80 by Lem1;
        then 2|^(42*4+1),80 are_congruent_mod 169 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 170;
        2|^10 = 170*6+4 & 4*4*4*4*2|^2 = 170*6+4 by Lm10,Lm2;
        then 2|^(10*4+2),4 are_congruent_mod 170 &
          4*4*4*4*2|^2 = 170*6+4 by Lem1;
        then 2|^(42*4+2),4 are_congruent_mod 170 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 171;
        2|^10 = 171*5+169 &
          169*169*169*169*2|^2 = 171*19081420+64 by Lm10,Lm2;
        then 2|^(10*4+2),64 are_congruent_mod 171 &
          64*64*64*64*2|^3 = 171*784898+170 by Lem1,Lm3;
        then 2|^(42*4+3),170 are_congruent_mod 171 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 172;
        2|^10 = 172*5+164 &
          164*164*164*164*2|^3 = 172*33646270+88 by Lm10,Lm3;
        then 2|^(10*4+3),88 are_congruent_mod 172 &
          88*88*88*88*2|^0 = 172*348660+16 by Lem1,Lm1;
        then 2|^(43*4+0),16 are_congruent_mod 172 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 173;
        hence thesis by XPRIMES1:173;
      end;
      suppose A: n = 174;
        2|^10 = 174*5+154 &
          154*154*154*154*2|^3 = 174*25859708+56 by Lm10,Lm3;
        then 2|^(10*4+3),56 are_congruent_mod 174 &
          56*56*56*56*2|^2 = 174*226080+64 by Lem1,Lm2;
        then 2|^(43*4+2),64 are_congruent_mod 174 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 175;
        2|^10 = 175*5+149 &
          149*149*149*149*2|^3 = 175*22531858+58 by Lm10,Lm3;
        then 2|^(10*4+3),58 are_congruent_mod 175 &
          58*58*58*58*2|^3 = 175*517325+93 by Lem1;
        then 2|^(43*4+3),93 are_congruent_mod 175 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 176;
        2|^11 = 176*11+112 & 112*112*112*112*2|^0 = 176*894045+16 by Lm11,Lm1;
        then 2|^(11*4+0),16 are_congruent_mod 176 &
          16*16*16*16*2|^0 = 176*372+64 by Lem1;
        then 2|^(44*4+0),64 are_congruent_mod 176 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 177;
        2|^11 = 177*11+101 &
          101*101*101*101*2|^0 = 177*587911+154 by Lm11,Lm1;
        then 2|^(11*4+0),154 are_congruent_mod 177 &
          154*154*154*154*2|^1 = 177*6355352+8 by Lem1;
        then 2|^(44*4+1),8 are_congruent_mod 177 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 178;
        2|^11 = 178*11+90 & 90*90*90*90*2|^0 = 178*368595+90 by Lm11,Lm1;
        then 2|^(11*4+0),90 are_congruent_mod 178 &
          90*90*90*90*2|^2 = 178*1474382+4 by Lem1,Lm2;
        then 2|^(44*4+2),4 are_congruent_mod 178 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 179;
        hence thesis by XPRIMES1:179;
      end;
      suppose A: n = 180;
        2|^11 = 180*11+68 & 68*68*68*68*2|^1 = 180*237570+152 by Lm11;
        then 2|^(11*4+1),152 are_congruent_mod 180 &
          152*152*152*152*2|^0 = 180*2965526+136 by Lem1,Lm1;
        then 2|^(45*4+0),136 are_congruent_mod 180 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 181;
        hence thesis by XPRIMES1:181;
      end;
      suppose A: n = 182;
        2|^11 = 182*11+46 & 46*46*46*46*2|^1 = 182*49202+148 by Lm11;
        then 2|^(11*4+1),148 are_congruent_mod 182 &
          148*148*148*148*2|^2 = 182*10544730+4 by Lem1,Lm2;
        then 2|^(45*4+2),4 are_congruent_mod 182 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 183;
        2|^11 = 183*11+35 & 35*35*35*35*2|^1 = 183*16400+50 by Lm11;
        then 2|^(11*4+1),50 are_congruent_mod 183 &
          50*50*50*50*2|^3 = 183*273224+8 by Lem1,Lm3;
        then 2|^(45*4+3),8 are_congruent_mod 183 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 184;
        2|^11 = 184*11+24 & 24*24*24*24*2|^2 = 184*7212+96 by Lm11,Lm2;
        then 2|^(11*4+2),96 are_congruent_mod 184 &
          96*96*96*96*2|^0 = 184*461601+72 by Lem1,Lm1;
        then 2|^(46*4+0),72 are_congruent_mod 184 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 185;
        2|^11 = 185*11+13 & 13*13*13*13*2|^2 = 185*617+99 by Lm11,Lm2;
        then 2|^(11*4+2),99 are_congruent_mod 185 &
          99*99*99*99*2|^1 = 185*1038482+32 by Lem1;
        then 2|^(46*4+1),32 are_congruent_mod 185 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 186;
        2|^11 = 186*11+2 & 2*2*2*2*2|^2 = 186*0+64 by Lm11,Lm2;
        then 2|^(11*4+2),64 are_congruent_mod 186 &
          64*64*64*64*2|^2 = 186*360800+64 by Lem1;
        then 2|^(46*4+2),64 are_congruent_mod 186 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 187;
        2|^11 = 187*10+178 &
          178*178*178*178*2|^2 = 187*21473280+64 by Lm11,Lm2;
        then 2|^(11*4+2),64 are_congruent_mod 187 &
          64*64*64*64*2|^3 = 187*717741+161 by Lem1,Lm3;
        then 2|^(46*4+3),161 are_congruent_mod 187 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 188;
        2|^11 = 188*10+168 &
          168*168*168*168*2|^3 = 188*33897624+96 by Lm11,Lm3;
        then 2|^(11*4+3),96 are_congruent_mod 188 &
          96*96*96*96*2|^0 = 188*451780+16 by Lem1,Lm1;
        then 2|^(47*4+0),16 are_congruent_mod 188 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 189;
        2|^11 = 189*10+158 &
          158*158*158*158*2|^3 = 189*26378890+158 by Lm11,Lm3;
        then 2|^(11*4+3),158 are_congruent_mod 189 &
          158*158*158*158*2|^1 = 189*6594722+134 by Lem1;
        then 2|^(47*4+1),134 are_congruent_mod 189 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 190;
        2|^11 = 190*10+148 &
          148*148*148*148*2|^3 = 190*20201482+148 by Lm11,Lm3;
        then 2|^(11*4+3),148 are_congruent_mod 190 &
          148*148*148*148*2|^2 = 190*10100741+74 by Lem1,Lm2;
        then 2|^(47*4+2),74 are_congruent_mod 190 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 191;
        hence thesis by XPRIMES1:191;
      end;
      suppose A: n = 192;
        2|^12 = 192*21+64 & 64*64*64*64*2|^0 = 192*87381+64 by Lm12,Lm1;
        then 2|^(12*4+0),64 are_congruent_mod 192 &
          64*64*64*64*2|^0 = 192*87381+64 by Lem1;
        then 2|^(48*4+0),64 are_congruent_mod 192 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 193;
        hence thesis by XPRIMES1:193;
      end;
      suppose A: n = 194;
        2|^12 = 194*21+22 & 22*22*22*22*2|^0 = 194*1207+98 by Lm12,Lm1;
        then 2|^(12*4+0),98 are_congruent_mod 194 &
          98*98*98*98*2|^2 = 194*1901790+4 by Lem1,Lm2;
        then 2|^(48*4+2),4 are_congruent_mod 194 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 195;
        2|^12 = 195*21+1 & 1*1*1*1*2|^0 = 195*0+1 by Lm12;
        then 2|^(12*4+0),1 are_congruent_mod 195 &
          1*1*1*1*2|^3 = 195*0+8 by Lem1,Lm3;
        then 2|^(48*4+3),8 are_congruent_mod 195 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 196;
        2|^12 = 196*20+176 &
          176*176*176*176*2|^1 = 196*9790944+128 by Lm12;
        then 2|^(12*4+1),128 are_congruent_mod 196 &
          128*128*128*128*2|^0 = 196*1369568+128 by Lem1,Lm1;
        then 2|^(49*4+0),128 are_congruent_mod 196 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 197;
        hence thesis by XPRIMES1:197;
      end;
      suppose A: n = 198;
        2|^12 = 198*20+136 &
          136*136*136*136*2|^1 = 198*3455575+182 by Lm12;
        then 2|^(12*4+1),182 are_congruent_mod 198 &
          182*182*182*182*2|^2 = 198*22165643+190 by Lem1,Lm2;
        then 2|^(49*4+2),190 are_congruent_mod 198 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 199;
        hence thesis by XPRIMES1:199;
      end;
      suppose A: n = 200;
        2|^12 = 200*20+96 & 96*96*96*96*2|^2 = 200*1698693+24 by Lm12,Lm2;
        then 2|^(12*4+2),24 are_congruent_mod 200 &
          24*24*24*24*2|^0 = 200*1658+176 by Lem1,Lm1;
        then 2|^(50*4+0),176 are_congruent_mod 200 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 201;
        2|^12 = 201*20+76 & 76*76*76*76*2|^2 = 201*663923+181 by Lm12,Lm2;
        then 2|^(12*4+2),181 are_congruent_mod 201 &
          181*181*181*181*2|^1 = 201*10679434+8 by Lem1;
        then 2|^(50*4+1),8 are_congruent_mod 201 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 202;
        2|^12 = 202*20+56 & 56*56*56*56*2|^2 = 202*194742+100 by Lm12,Lm2;
        then 2|^(12*4+2),100 are_congruent_mod 202 &
          100*100*100*100*2|^2 = 202*1980198+4 by Lem1;
        then 2|^(50*4+2),4 are_congruent_mod 202 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 203;
        2|^12 = 203*20+36 & 36*36*36*36*2|^2 = 203*33095+179 by Lm12,Lm2;
        then 2|^(12*4+2),179 are_congruent_mod 203 &
          179*179*179*179*2|^3 = 203*40458154+186 by Lem1,Lm3;
        then 2|^(50*4+3),186 are_congruent_mod 203 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 204;
        2|^12 = 204*20+16 & 16*16*16*16*2|^3 = 204*2570+8 by Lm12,Lm3;
        then 2|^(12*4+3),8 are_congruent_mod 204 &
          8*8*8*8*2|^0 = 204*20+16 by Lem1,Lm1;
        then 2|^(51*4+0),16 are_congruent_mod 204 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 204 < n & n <= 304;
      then 204+1 <= n & n <= 304 by NAT_1:13;
      then n = 205 or ... or n = 304;
      then per cases;
      suppose A: n = 205;
        2|^12 = 205*19+201 &
          201*201*201*201*2|^3 = 205*63697201+203 by Lm12,Lm3;
        then 2|^(12*4+3),203 are_congruent_mod 205 &
          203*203*203*203*2|^1 = 205*16567626+32 by Lem1;
        then 2|^(51*4+1),32 are_congruent_mod 205 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 206;
        2|^12 = 206*19+182 &
          182*182*182*182*2|^3 = 206*42609684+104 by Lm12,Lm3;
        then 2|^(12*4+3),104 are_congruent_mod 206 &
          104*104*104*104*2|^2 = 206*2271570+4 by Lem1,Lm2;
        then 2|^(51*4+2),4 are_congruent_mod 206 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 207;
        2|^12 = 207*19+163 &
          163*163*163*163*2|^3 = 207*27281613+197 by Lm12,Lm3;
        then 2|^(12*4+3),197 are_congruent_mod 207 &
          197*197*197*197*2|^3 = 207*58208250+98 by Lem1;
        then 2|^(51*4+3),98 are_congruent_mod 207 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 208;
        2|^13 = 208*39+80 & 80*80*80*80*2|^0 = 208*196923+16 by Lm13,Lm1;
        then 2|^(13*4+0),16 are_congruent_mod 208 &
          16*16*16*16*2|^0 = 208*315+16 by Lem1;
        then 2|^(52*4+0),16 are_congruent_mod 208 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 209;
        2|^13 = 209*39+41 & 41*41*41*41*2|^0 = 209*13520+81 by Lm13,Lm1;
        then 2|^(13*4+0),81 are_congruent_mod 209 &
          81*81*81*81*2|^1 = 209*411930+72 by Lem1;
        then 2|^(52*4+1),72 are_congruent_mod 209 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 210;
        2|^13 = 210*39+2 & 2*2*2*2*2|^0 = 210*0+16 by Lm13,Lm1;
        then 2|^(13*4+0),16 are_congruent_mod 210 &
          16*16*16*16*2|^2 = 210*1248+64 by Lem1,Lm2;
        then 2|^(52*4+2),64 are_congruent_mod 210 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 211;
        hence thesis by XPRIMES1:211;
      end;
      suppose A: n = 212;
        2|^13 = 212*38+136 &
          136*136*136*136*2|^1 = 212*3227377+108 by Lm13;
        then 2|^(13*4+1),108 are_congruent_mod 212 &
          108*108*108*108*2|^0 = 212*641740+16 by Lem1,Lm1;
        then 2|^(53*4+0),16 are_congruent_mod 212 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 213;
        2|^13 = 213*38+98 & 98*98*98*98*2|^1 = 213*866073+83 by Lm13;
        then 2|^(13*4+1),83 are_congruent_mod 213 &
          83*83*83*83*2|^1 = 213*445618+8 by Lem1;
        then 2|^(53*4+1),8 are_congruent_mod 213 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 214;
        2|^13 = 214*38+60 & 60*60*60*60*2|^1 = 214*121121+106 by Lm13;
        then 2|^(13*4+1),106 are_congruent_mod 214 &
          106*106*106*106*2|^2 = 214*2359770+4 by Lem1,Lm2;
        then 2|^(53*4+2),4 are_congruent_mod 214 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 215;
        2|^13 = 215*38+22 & 22*22*22*22*2|^1 = 215*2179+27 by Lm13;
        then 2|^(13*4+1),27 are_congruent_mod 215 &
          27*27*27*27*2|^3 = 215*19774+118 by Lem1,Lm3;
        then 2|^(53*4+3),118 are_congruent_mod 215 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 216;
        2|^13 = 216*37+200 &
          200*200*200*200*2|^2 = 216*29629629+136 by Lm13,Lm2;
        then 2|^(13*4+2),136 are_congruent_mod 216 &
          136*136*136*136*2|^0 = 216*1583805+136 by Lem1,Lm1;
        then 2|^(54*4+0),136 are_congruent_mod 216 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 217;
        2|^13 = 217*37+163 &
          163*163*163*163*2|^2 = 217*13012198+78 by Lm13,Lm2;
        then 2|^(13*4+2),78 are_congruent_mod 217 &
          78*78*78*78*2|^1 = 217*341152+128 by Lem1;
        then 2|^(54*4+1),128 are_congruent_mod 217 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 218;
        2|^13 = 218*37+126 &
          126*126*126*126*2|^2 = 218*4624722+108 by Lm13,Lm2;
        then 2|^(13*4+2),108 are_congruent_mod 218 &
          108*108*108*108*2|^2 = 218*2496310+4 by Lem1;
        then 2|^(54*4+2),4 are_congruent_mod 218 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 219;
        2|^13 = 219*37+89 & 89*89*89*89*2|^2 = 219*1145977+1 by Lm13,Lm2;
        then 2|^(13*4+2),1 are_congruent_mod 219 &
          1*1*1*1*2|^3 = 219*0+8 by Lem1,Lm3;
        then 2|^(54*4+3),8 are_congruent_mod 219 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 220;
        2|^13 = 220*37+52 & 52*52*52*52*2|^3 = 220*265876+208 by Lm13,Lm3;
        then 2|^(13*4+3),208 are_congruent_mod 220 &
          208*208*208*208*2|^0 = 220*8508062+56 by Lem1,Lm1;
        then 2|^(55*4+0),56 are_congruent_mod 220 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 221;
        2|^13 = 221*37+15 & 15*15*15*15*2|^3 = 221*1832+128 by Lm13,Lm3;
        then 2|^(13*4+3),128 are_congruent_mod 221 &
          128*128*128*128*2|^1 = 221*2429280+32 by Lem1;
        then 2|^(55*4+1),32 are_congruent_mod 221 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 222;
        2|^13 = 222*36+200 &
          200*200*200*200*2|^3 = 222*57657657+146 by Lm13,Lm3;
        then 2|^(13*4+3),146 are_congruent_mod 222 &
          146*146*146*146*2|^2 = 222*8186880+64 by Lem1,Lm2;
        then 2|^(55*4+2),64 are_congruent_mod 222 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 223;
        hence thesis by XPRIMES1:223;
      end;
      suppose A: n = 224;
        2|^14 = 224*73+32 & 32*32*32*32*2|^0 = 224*4681+32 by Lm14,Lm1;
        then 2|^(14*4+0),32 are_congruent_mod 224 &
          32*32*32*32*2|^0 = 224*4681+32 by Lem1;
        then 2|^(56*4+0),32 are_congruent_mod 224 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 225;
        2|^14 = 225*72+184 &
          184*184*184*184*2|^0 = 225*5094349+211 by Lm14,Lm1;
        then 2|^(14*4+0),211 are_congruent_mod 225 &
          211*211*211*211*2|^1 = 225*17618839+107 by Lem1;
        then 2|^(56*4+1),107 are_congruent_mod 225 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 226;
        2|^14 = 226*72+112 &
          112*112*112*112*2|^0 = 226*696247+114 by Lm14,Lm1;
        then 2|^(14*4+0),114 are_congruent_mod 226 &
          114*114*114*114*2|^2 = 226*2989310+4 by Lem1,Lm2;
        then 2|^(56*4+2),4 are_congruent_mod 226 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 227;
        hence thesis by XPRIMES1:227;
      end;
      suppose A: n = 228;
        2|^14 = 228*71+196 &
          196*196*196*196*2|^1 = 228*12945518+8 by Lm14;
        then 2|^(14*4+1),8 are_congruent_mod 228 &
          8*8*8*8*2|^0 = 228*17+220 by Lem1,Lm1;
        then 2|^(57*4+0),220 are_congruent_mod 228 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 229;
        hence thesis by XPRIMES1:229;
      end;
      suppose A: n = 230;
        2|^14 = 230*71+54 & 54*54*54*54*2|^1 = 230*73939+142 by Lm14;
        then 2|^(14*4+1),142 are_congruent_mod 230 &
          142*142*142*142*2|^2 = 230*7071076+104 by Lem1,Lm2;
        then 2|^(57*4+2),104 are_congruent_mod 230 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 231;
        2|^14 = 231*70+214 &
          214*214*214*214*2|^1 = 231*18158213+29 by Lm14;
        then 2|^(14*4+1),29 are_congruent_mod 231 &
          29*29*29*29*2|^3 = 231*24494+134 by Lem1,Lm3;
        then 2|^(57*4+3),134 are_congruent_mod 231 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 232;
        2|^14 = 232*70+144 &
          144*144*144*144*2|^2 = 232*7413477+120 by Lm14,Lm2;
        then 2|^(14*4+2),120 are_congruent_mod 232 &
          120*120*120*120*2|^0 = 232*893793+24 by Lem1,Lm1;
        then 2|^(58*4+0),24 are_congruent_mod 232 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 233;
        hence thesis by XPRIMES1:233;
      end;
      suppose A: n = 234;
        2|^14 = 234*70+4 & 4*4*4*4*2|^2 = 234*4+88 by Lm14,Lm2;
        then 2|^(14*4+2),88 are_congruent_mod 234 &
          88*88*88*88*2|^2 = 234*1025120+64 by Lem1;
        then 2|^(58*4+2),64 are_congruent_mod 234 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 235;
        2|^14 = 235*69+169 &
          169*169*169*169*2|^2 = 235*13884778+54 by Lm14,Lm2;
        then 2|^(14*4+2),54 are_congruent_mod 235 &
          54*54*54*54*2|^3 = 235*289465+173 by Lem1,Lm3;
        then 2|^(58*4+3),173 are_congruent_mod 235 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 236;
        2|^14 = 236*69+100 &
          100*100*100*100*2|^3 = 236*3389830+120 by Lm14,Lm3;
        then 2|^(14*4+3),120 are_congruent_mod 236 &
          120*120*120*120*2|^0 = 236*878644+16 by Lem1,Lm1;
        then 2|^(59*4+0),16 are_congruent_mod 236 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 237;
        2|^14 = 237*69+31 & 31*31*31*31*2|^3 = 237*31173+167 by Lm14,Lm3;
        then 2|^(14*4+3),167 are_congruent_mod 237 &
          167*167*167*167*2|^1 = 237*6563682+8 by Lem1;
        then 2|^(59*4+1),8 are_congruent_mod 237 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 238;
        2|^14 = 238*68+200 &
          200*200*200*200*2|^3 = 238*53781512+144 by Lm14,Lm3;
        then 2|^(14*4+3),144 are_congruent_mod 238 &
          144*144*144*144*2|^2 = 238*7226583+30 by Lem1,Lm2;
        then 2|^(59*4+2),30 are_congruent_mod 238 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 239;
        hence thesis by XPRIMES1:239;
      end;
      suppose A: n = 240;
        2|^15 = 240*136+128 &
          128*128*128*128*2|^0 = 240*1118481+16 by Lm15,Lm1;
        then 2|^(15*4+0),16 are_congruent_mod 240 &
          16*16*16*16*2|^0 = 240*273+16 by Lem1;
        then 2|^(60*4+0),16 are_congruent_mod 240 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 241;
        hence thesis by XPRIMES1:241;
      end;
      suppose A: n = 242;
        2|^15 = 242*135+98 & 98*98*98*98*2|^0 = 242*381143+210 by Lm15,Lm1;
        then 2|^(15*4+0),210 are_congruent_mod 242 &
          210*210*210*210*2|^2 = 242*32145619+202 by Lem1,Lm2;
        then 2|^(60*4+2),202 are_congruent_mod 242 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 243;
        2|^15 = 243*134+206 &
          206*206*206*206*2|^0 = 243*7410757+145 by Lm15,Lm1;
        then 2|^(15*4+0),145 are_congruent_mod 243 &
          145*145*145*145*2|^3 = 243*14553106+242 by Lem1,Lm3;
        then 2|^(60*4+3),242 are_congruent_mod 243 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 244;
        2|^15 = 244*134+72 & 72*72*72*72*2|^1 = 244*220277+124 by Lm15;
        then 2|^(15*4+1),124 are_congruent_mod 244 &
          124*124*124*124*2|^0 = 244*968940+16 by Lem1,Lm1;
        then 2|^(61*4+0),16 are_congruent_mod 244 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 245;
        2|^15 = 245*133+183 &
          183*183*183*183*2|^1 = 245*9155209+37 by Lm15;
        then 2|^(15*4+1),37 are_congruent_mod 245 &
          37*37*37*37*2|^1 = 245*15299+67 by Lem1;
        then 2|^(61*4+1),67 are_congruent_mod 245 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 246;
        2|^15 = 246*133+50 & 50*50*50*50*2|^1 = 246*50813+2 by Lm15;
        then 2|^(15*4+1),2 are_congruent_mod 246 &
          2*2*2*2*2|^2 = 246*0+64 by Lem1,Lm2;
        then 2|^(61*4+2),64 are_congruent_mod 246 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 247;
        2|^15 = 247*132+164 &
          164*164*164*164*2|^1 = 247*5857447+223 by Lm15;
        then 2|^(15*4+1),223 are_congruent_mod 247 &
          223*223*223*223*2|^3 = 247*80096305+193 by Lem1,Lm3;
        then 2|^(61*4+3),193 are_congruent_mod 247 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 248;
        2|^15 = 248*132+32 & 32*32*32*32*2|^2 = 248*16912+128 by Lm15,Lm2;
        then 2|^(15*4+2),128 are_congruent_mod 248 &
          128*128*128*128*2|^0 = 248*1082401+8 by Lem1,Lm1;
        then 2|^(62*4+0),8 are_congruent_mod 248 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 249;
        2|^15 = 249*131+149 &
          149*149*149*149*2|^2 = 249*7917821+175 by Lm15,Lm2;
        then 2|^(15*4+2),175 are_congruent_mod 249 &
          175*175*175*175*2|^1 = 249*7533258+8 by Lem1;
        then 2|^(62*4+1),8 are_congruent_mod 249 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 250;
        2|^15 = 250*131+18 & 18*18*18*18*2|^2 = 250*1679+154 by Lm15,Lm2;
        then 2|^(15*4+2),154 are_congruent_mod 250 &
          154*154*154*154*2|^2 = 250*8999178+124 by Lem1;
        then 2|^(62*4+2),124 are_congruent_mod 250 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 251;
        hence thesis by XPRIMES1:251;
      end;
      suppose A: n = 252;
        2|^15 = 252*130+8 & 8*8*8*8*2|^3 = 252*130+8 by Lm15,Lm3;
        then 2|^(15*4+3),8 are_congruent_mod 252 &
          8*8*8*8*2|^0 = 252*16+64 by Lem1,Lm1;
        then 2|^(63*4+0),64 are_congruent_mod 252 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 253;
        2|^15 = 253*129+131 &
          131*131*131*131*2|^3 = 253*9312250+118 by Lm15,Lm3;
        then 2|^(15*4+3),118 are_congruent_mod 253 &
          118*118*118*118*2|^1 = 253*1532630+162 by Lem1;
        then 2|^(63*4+1),162 are_congruent_mod 253 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 254;
        2|^15 = 254*129+2 & 2*2*2*2*2|^3 = 254*0+128 by Lm15,Lm3;
        then 2|^(15*4+3),128 are_congruent_mod 254 &
          128*128*128*128*2|^2 = 254*4227330+4 by Lem1,Lm2;
        then 2|^(63*4+2),4 are_congruent_mod 254 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 255;
        2|^15 = 255*128+128 &
          128*128*128*128*2|^3 = 255*8421504+128 by Lm15,Lm3;
        then 2|^(15*4+3),128 are_congruent_mod 255 &
          128*128*128*128*2|^3 = 255*8421504+128 by Lem1;
        then 2|^(63*4+3),128 are_congruent_mod 255 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 256;
        2|^16 = 256*256+0 & 0*0*0*0*2|^0 = 256*0+0 by Lm16;
        then 2|^(16*4+0),0 are_congruent_mod 256 &
          0*0*0*0*2|^0 = 256*0+0 by Lem1;
        then 2|^(64*4+0),0 are_congruent_mod 256 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 257;
        hence thesis by XPRIMES1:257;
      end;
      suppose A: n = 258;
        2|^16 = 258*254+4 & 4*4*4*4*2|^0 = 258*0+256 by Lm16,Lm1;
        then 2|^(16*4+0),256 are_congruent_mod 258 &
          256*256*256*256*2|^2 = 258*66588640+64 by Lem1,Lm2;
        then 2|^(64*4+2),64 are_congruent_mod 258 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 259;
        2|^16 = 259*253+9 & 9*9*9*9*2|^0 = 259*25+86 by Lm16,Lm1;
        then 2|^(16*4+0),86 are_congruent_mod 259 &
          86*86*86*86*2|^3 = 259*1689600+128 by Lem1,Lm3;
        then 2|^(64*4+3),128 are_congruent_mod 259 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 260;
        2|^16 = 260*252+16 & 16*16*16*16*2|^1 = 260*504+32 by Lm16;
        then 2|^(16*4+1),32 are_congruent_mod 260 &
          32*32*32*32*2|^0 = 260*4032+256 by Lem1,Lm1;
        then 2|^(65*4+0),256 are_congruent_mod 260 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 261;
        2|^16 = 261*251+25 & 25*25*25*25*2|^1 = 261*2993+77 by Lm16;
        then 2|^(16*4+1),77 are_congruent_mod 261 &
          77*77*77*77*2|^1 = 261*269371+251 by Lem1;
        then 2|^(65*4+1),251 are_congruent_mod 261 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 262;
        2|^16 = 262*250+36 & 36*36*36*36*2|^1 = 262*12821+130 by Lm16;
        then 2|^(16*4+1),130 are_congruent_mod 262 &
          130*130*130*130*2|^2 = 262*4360458+4 by Lem1,Lm2;
        then 2|^(65*4+2),4 are_congruent_mod 262 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 263;
        hence thesis by XPRIMES1:263;
      end;
      suppose A: n = 264;
        2|^16 = 264*248+64 & 64*64*64*64*2|^2 = 264*254200+64 by Lm16,Lm2;
        then 2|^(16*4+2),64 are_congruent_mod 264 &
          64*64*64*64*2|^0 = 264*63550+16 by Lem1,Lm1;
        then 2|^(66*4+0),16 are_congruent_mod 264 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 265;
        2|^16 = 265*247+81 & 81*81*81*81*2|^2 = 265*649761+219 by Lm16,Lm2;
        then 2|^(16*4+2),219 are_congruent_mod 265 &
          219*219*219*219*2|^1 = 265*17360434+32 by Lem1;
        then 2|^(66*4+1),32 are_congruent_mod 265 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 266;
        2|^16 = 266*246+100 &
          100*100*100*100*2|^2 = 266*1503759+106 by Lm16,Lm2;
        then 2|^(16*4+2),106 are_congruent_mod 266 &
          106*106*106*106*2|^2 = 266*1898461+158 by Lem1;
        then 2|^(66*4+2),158 are_congruent_mod 266 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 267;
        2|^16 = 267*245+121 &
          121*121*121*121*2|^2 = 267*3211369+1 by Lm16,Lm2;
        then 2|^(16*4+2),1 are_congruent_mod 267 &
          1*1*1*1*2|^3 = 267*0+8 by Lem1,Lm3;
        then 2|^(66*4+3),8 are_congruent_mod 267 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 268;
        2|^16 = 268*244+144 &
          144*144*144*144*2|^3 = 268*12835274+136 by Lm16,Lm3;
        then 2|^(16*4+3),136 are_congruent_mod 268 &
          136*136*136*136*2|^0 = 268*1276500+16 by Lem1,Lm1;
        then 2|^(67*4+0),16 are_congruent_mod 268 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 269;
        hence thesis by XPRIMES1:269;
      end;
      suppose A: n = 270;
        2|^16 = 270*242+196 &
          196*196*196*196*2|^3 = 270*43727083+38 by Lm16,Lm3;
        then 2|^(16*4+3),38 are_congruent_mod 270 &
          38*38*38*38*2|^2 = 270*30890+244 by Lem1,Lm2;
        then 2|^(67*4+2),244 are_congruent_mod 270 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 271;
        hence thesis by XPRIMES1:271;
      end;
      suppose A: n = 272;
        2|^4 = 272*0+16 & 16*16*16*16*2|^1 = 272*481+240 by Lm4;
        then 2|^(4*4+1),240 are_congruent_mod 272 &
          240*240*240*240*2|^0 = 272*12197647+16 by Lem1,Lm1;
        then 2|^(17*4+0),16 are_congruent_mod 272 &
          16*16*16*16*2|^0 = 272*240+256 by LemS;
        then 2|^(68*4+0),256 are_congruent_mod 272 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 273;
        2|^4 = 273*0+16 & 16*16*16*16*2|^1 = 273*480+32 by Lm4;
        then 2|^(4*4+1),32 are_congruent_mod 273 &
          32*32*32*32*2|^0 = 273*3840+256 by Lem1,Lm1;
        then 2|^(17*4+0),256 are_congruent_mod 273 &
          256*256*256*256*2|^1 = 273*31464961+239 by LemS;
        then 2|^(68*4+1),239 are_congruent_mod 273 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 274;
        2|^4 = 274*0+16 & 16*16*16*16*2|^1 = 274*478+100 by Lm4;
        then 2|^(4*4+1),100 are_congruent_mod 274 &
          100*100*100*100*2|^0 = 274*364963+138 by Lem1,Lm1;
        then 2|^(17*4+0),138 are_congruent_mod 274 &
          138*138*138*138*2|^2 = 274*5294510+4 by LemS,Lm2;
        then 2|^(68*4+2),4 are_congruent_mod 274 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 275;
        2|^4 = 275*0+16 & 16*16*16*16*2|^1 = 275*476+172 by Lm4;
        then 2|^(4*4+1),172 are_congruent_mod 275 &
          172*172*172*172*2|^0 = 275*3182592+256 by Lem1,Lm1;
        then 2|^(17*4+0),256 are_congruent_mod 275 &
          256*256*256*256*2|^3 = 275*124944503+43 by LemS,Lm3;
        then 2|^(68*4+3),43 are_congruent_mod 275 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 276;
        2|^4 = 276*0+16 & 16*16*16*16*2|^1 = 276*474+248 by Lm4;
        then 2|^(4*4+1),248 are_congruent_mod 276 &
          248*248*248*248*2|^1 = 276*27411174+8 by Lem1;
        then 2|^(17*4+1),8 are_congruent_mod 276 &
          8*8*8*8*2|^0 = 276*14+232 by LemS,Lm1;
        then 2|^(69*4+0),232 are_congruent_mod 276 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 277;
        hence thesis by XPRIMES1:277;
      end;
      suppose A: n = 278;
        2|^4 = 278*0+16 & 16*16*16*16*2|^1 = 278*471+134 by Lm4;
        then 2|^(4*4+1),134 are_congruent_mod 278 &
          134*134*134*134*2|^1 = 278*2319553+138 by Lem1;
        then 2|^(17*4+1),138 are_congruent_mod 278 &
          138*138*138*138*2|^2 = 278*5218330+4 by LemS,Lm2;
        then 2|^(69*4+2),4 are_congruent_mod 278 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 279;
        2|^4 = 279*0+16 & 16*16*16*16*2|^1 = 279*469+221 by Lm4;
        then 2|^(4*4+1),221 are_congruent_mod 279 &
          221*221*221*221*2|^1 = 279*17099951+233 by Lem1;
        then 2|^(17*4+1),233 are_congruent_mod 279 &
          233*233*233*233*2|^3 = 279*84510265+233 by LemS,Lm3;
        then 2|^(69*4+3),233 are_congruent_mod 279 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 280;
        2|^4 = 280*0+16 & 16*16*16*16*2|^1 = 280*468+32 by Lm4;
        then 2|^(4*4+1),32 are_congruent_mod 280 &
          32*32*32*32*2|^2 = 280*14979+184 by Lem1,Lm2;
        then 2|^(17*4+2),184 are_congruent_mod 280 &
          184*184*184*184*2|^0 = 280*4093674+16 by LemS,Lm1;
        then 2|^(70*4+0),16 are_congruent_mod 280 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 281;
        hence thesis by XPRIMES1:281;
      end;
      suppose A: n = 282;
        2|^4 = 282*0+16 & 16*16*16*16*2|^1 = 282*464+224 by Lm4;
        then 2|^(4*4+1),224 are_congruent_mod 282 &
          224*224*224*224*2|^2 = 282*35711077+190 by Lem1,Lm2;
        then 2|^(17*4+2),190 are_congruent_mod 282 &
          190*190*190*190*2|^2 = 282*18485248+64 by LemS;
        then 2|^(70*4+2),64 are_congruent_mod 282 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 283;
        hence thesis by XPRIMES1:283;
      end;
      suppose A: n = 284;
        2|^4 = 284*0+16 & 16*16*16*16*2|^1 = 284*461+148 by Lm4;
        then 2|^(4*4+1),148 are_congruent_mod 284 &
          148*148*148*148*2|^3 = 284*13515076+144 by Lem1,Lm3;
        then 2|^(17*4+3),144 are_congruent_mod 284 &
          144*144*144*144*2|^0 = 284*1514020+16 by LemS,Lm1;
        then 2|^(71*4+0),16 are_congruent_mod 284 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 285;
        2|^4 = 285*0+16 & 16*16*16*16*2|^1 = 285*459+257 by Lm4;
        then 2|^(4*4+1),257 are_congruent_mod 285 &
          257*257*257*257*2|^3 = 285*122455309+143 by Lem1,Lm3;
        then 2|^(17*4+3),143 are_congruent_mod 285 &
          143*143*143*143*2|^1 = 285*2934467+107 by LemS;
        then 2|^(71*4+1),107 are_congruent_mod 285 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 286;
        2|^4 = 286*0+16 & 16*16*16*16*2|^1 = 286*458+84 by Lm4;
        then 2|^(4*4+1),84 are_congruent_mod 286 &
          84*84*84*84*2|^3 = 286*1392647+46 by Lem1,Lm3;
        then 2|^(17*4+3),46 are_congruent_mod 286 &
          46*46*46*46*2|^2 = 286*62621+218 by LemS,Lm2;
        then 2|^(71*4+2),218 are_congruent_mod 286 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 287;
        2|^4 = 287*0+16 & 16*16*16*16*2|^1 = 287*456+200 by Lm4;
        then 2|^(4*4+1),200 are_congruent_mod 287 &
          200*200*200*200*2|^3 = 287*44599303+39 by Lem1,Lm3;
        then 2|^(17*4+3),39 are_congruent_mod 287 &
          39*39*39*39*2|^3 = 287*64486+46 by LemS;
        then 2|^(71*4+3),46 are_congruent_mod 287 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 288;
        2|^4 = 288*0+16 & 16*16*16*16*2|^2 = 288*910+64 by Lm4,Lm2;
        then 2|^(4*4+2),64 are_congruent_mod 288 &
          64*64*64*64*2|^0 = 288*58254+64 by Lem1,Lm1;
        then 2|^(18*4+0),64 are_congruent_mod 288 &
          64*64*64*64*2|^0 = 288*58254+64 by LemS;
        then 2|^(72*4+0),64 are_congruent_mod 288 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 289;
        2|^4 = 289*0+16 & 16*16*16*16*2|^2 = 289*907+21 by Lm4,Lm2;
        then 2|^(4*4+2),21 are_congruent_mod 289 &
          21*21*21*21*2|^0 = 289*672+273 by Lem1,Lm1;
        then 2|^(18*4+0),273 are_congruent_mod 289 &
          273*273*273*273*2|^1 = 289*38439943+155 by LemS;
        then 2|^(72*4+1),155 are_congruent_mod 289 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 290;
        2|^4 = 290*0+16 & 16*16*16*16*2|^2 = 290*903+274 by Lm4,Lm2;
        then 2|^(4*4+2),274 are_congruent_mod 290 &
          274*274*274*274*2|^0 = 290*19435881+286 by Lem1,Lm1;
        then 2|^(18*4+0),286 are_congruent_mod 290 &
          286*286*286*286*2|^2 = 290*92283939+154 by LemS,Lm2;
        then 2|^(72*4+2),154 are_congruent_mod 290 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 291;
        2|^4 = 291*0+16 & 16*16*16*16*2|^2 = 291*900+244 by Lm4,Lm2;
        then 2|^(4*4+2),244 are_congruent_mod 291 &
          244*244*244*244*2|^0 = 291*12180533+193 by Lem1,Lm1;
        then 2|^(18*4+0),193 are_congruent_mod 291 &
          193*193*193*193*2|^3 = 291*38144000+8 by LemS,Lm3;
        then 2|^(72*4+3),8 are_congruent_mod 291 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 292;
        2|^4 = 292*0+16 & 16*16*16*16*2|^2 = 292*897+220 by Lm4,Lm2;
        then 2|^(4*4+2),220 are_congruent_mod 292 &
          220*220*220*220*2|^1 = 292*16044931+148 by Lem1;
        then 2|^(18*4+1),148 are_congruent_mod 292 &
          148*148*148*148*2|^0 = 292*1643100+16 by LemS,Lm1;
        then 2|^(73*4+0),16 are_congruent_mod 292 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 293;
        hence thesis by XPRIMES1:293;
      end;
      suppose A: n = 294;
        2|^4 = 294*0+16 & 16*16*16*16*2|^2 = 294*891+190 by Lm4,Lm2;
        then 2|^(4*4+2),190 are_congruent_mod 294 &
          190*190*190*190*2|^1 = 294*8865374+44 by Lem1;
        then 2|^(18*4+1),44 are_congruent_mod 294 &
          44*44*44*44*2|^2 = 294*50994+148 by LemS,Lm2;
        then 2|^(73*4+2),148 are_congruent_mod 294 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 295;
        2|^4 = 295*0+16 & 16*16*16*16*2|^2 = 295*888+184 by Lm4,Lm2;
        then 2|^(4*4+2),184 are_congruent_mod 295 &
          184*184*184*184*2|^1 = 295*7771042+82 by Lem1;
        then 2|^(18*4+1),82 are_congruent_mod 295 &
          82*82*82*82*2|^3 = 295*1226092+268 by LemS,Lm3;
        then 2|^(73*4+3),268 are_congruent_mod 295 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 296;
        2|^4 = 296*0+16 & 16*16*16*16*2|^2 = 296*885+184 by Lm4,Lm2;
        then 2|^(4*4+2),184 are_congruent_mod 296 &
          184*184*184*184*2|^2 = 296*15489577+152 by Lem1;
        then 2|^(18*4+2),152 are_congruent_mod 296 &
          152*152*152*152*2|^0 = 296*1803360+256 by LemS,Lm1;
        then 2|^(74*4+0),256 are_congruent_mod 296 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 297;
        2|^4 = 297*0+16 & 16*16*16*16*2|^2 = 297*882+190 by Lm4,Lm2;
        then 2|^(4*4+2),190 are_congruent_mod 297 &
          190*190*190*190*2|^2 = 297*17551649+247 by Lem1;
        then 2|^(18*4+2),247 are_congruent_mod 297 &
          247*247*247*247*2|^1 = 297*25064633+161 by LemS;
        then 2|^(74*4+1),161 are_congruent_mod 297 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 298;
        2|^4 = 298*0+16 & 16*16*16*16*2|^2 = 298*879+202 by Lm4,Lm2;
        then 2|^(4*4+2),202 are_congruent_mod 298 &
          202*202*202*202*2|^2 = 298*22348542+148 by Lem1;
        then 2|^(18*4+2),148 are_congruent_mod 298 &
          148*148*148*148*2|^2 = 298*6440070+4 by LemS;
        then 2|^(74*4+2),4 are_congruent_mod 298 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 299;
        2|^4 = 299*0+16 & 16*16*16*16*2|^2 = 299*876+220 by Lm4,Lm2;
        then 2|^(4*4+2),220 are_congruent_mod 299 &
          220*220*220*220*2|^2 = 299*31338595+95 by Lem1;
        then 2|^(18*4+2),95 are_congruent_mod 299 &
          95*95*95*95*2|^3 = 299*2179280+280 by LemS,Lm3;
        then 2|^(74*4+3),280 are_congruent_mod 299 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 300;
        2|^4 = 300*0+16 & 16*16*16*16*2|^2 = 300*873+244 by Lm4,Lm2;
        then 2|^(4*4+2),244 are_congruent_mod 300 &
          244*244*244*244*2|^3 = 300*94520941+68 by Lem1,Lm3;
        then 2|^(18*4+3),68 are_congruent_mod 300 &
          68*68*68*68*2|^0 = 300*71271+76 by LemS,Lm1;
        then 2|^(75*4+0),76 are_congruent_mod 300 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 301;
        2|^4 = 301*0+16 & 16*16*16*16*2|^2 = 301*870+274 by Lm4,Lm2;
        then 2|^(4*4+2),274 are_congruent_mod 301 &
          274*274*274*274*2|^3 = 301*149804804+204 by Lem1,Lm3;
        then 2|^(18*4+3),204 are_congruent_mod 301 &
          204*204*204*204*2|^1 = 301*11507584+128 by LemS;
        then 2|^(75*4+1),128 are_congruent_mod 301 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 302;
        2|^4 = 302*0+16 & 16*16*16*16*2|^2 = 302*868+8 by Lm4,Lm2;
        then 2|^(4*4+2),8 are_congruent_mod 302 &
          8*8*8*8*2|^3 = 302*108+152 by Lem1,Lm3;
        then 2|^(18*4+3),152 are_congruent_mod 302 &
          152*152*152*152*2|^2 = 302*7070130+4 by LemS,Lm2;
        then 2|^(75*4+2),4 are_congruent_mod 302 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 303;
        2|^4 = 303*0+16 & 16*16*16*16*2|^2 = 303*865+49 by Lm4,Lm2;
        then 2|^(4*4+2),49 are_congruent_mod 303 &
          49*49*49*49*2|^3 = 303*152205+293 by Lem1,Lm3;
        then 2|^(18*4+3),293 are_congruent_mod 303 &
          293*293*293*293*2|^3 = 303*194588800+8 by LemS;
        then 2|^(75*4+3),8 are_congruent_mod 303 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 304;
        2|^4 = 304*0+16 & 16*16*16*16*2|^3 = 304*1724+192 by Lm4,Lm3;
        then 2|^(4*4+3),192 are_congruent_mod 304 &
          192*192*192*192*2|^0 = 304*4470245+16 by Lem1,Lm1;
        then 2|^(19*4+0),16 are_congruent_mod 304 &
          16*16*16*16*2|^0 = 304*215+176 by LemS;
        then 2|^(76*4+0),176 are_congruent_mod 304 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 304 < n & n <= 404;
      then 304+1 <= n & n <= 404 by NAT_1:13;
      then n = 305 or ... or n = 404;
      then per cases;
      suppose A: n = 305;
        2|^4 = 305*0+16 & 16*16*16*16*2|^3 = 305*1718+298 by Lm4,Lm3;
        then 2|^(4*4+3),298 are_congruent_mod 305 &
          298*298*298*298*2|^0 = 305*25856230+266 by Lem1,Lm1;
        then 2|^(19*4+0),266 are_congruent_mod 305 &
          266*266*266*266*2|^1 = 305*32828928+32 by LemS;
        then 2|^(76*4+1),32 are_congruent_mod 305 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 306;
        2|^4 = 306*0+16 & 16*16*16*16*2|^3 = 306*1713+110 by Lm4,Lm3;
        then 2|^(4*4+3),110 are_congruent_mod 306 &
          110*110*110*110*2|^0 = 306*478464+16 by Lem1,Lm1;
        then 2|^(19*4+0),16 are_congruent_mod 306 &
          16*16*16*16*2|^2 = 306*856+208 by LemS,Lm2;
        then 2|^(76*4+2),208 are_congruent_mod 306 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 307;
        hence thesis by XPRIMES1:307;
      end;
      suppose A: n = 308;
        2|^4 = 308*0+16 & 16*16*16*16*2|^3 = 308*1702+72 by Lm4,Lm3;
        then 2|^(4*4+3),72 are_congruent_mod 308 &
          72*72*72*72*2|^1 = 308*174505+172 by Lem1;
        then 2|^(19*4+1),172 are_congruent_mod 308 &
          172*172*172*172*2|^0 = 308*2841600+256 by LemS,Lm1;
        then 2|^(77*4+0),256 are_congruent_mod 308 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 309;
        2|^4 = 309*0+16 & 16*16*16*16*2|^3 = 309*1696+224 by Lm4,Lm3;
        then 2|^(4*4+3),224 are_congruent_mod 309 &
          224*224*224*224*2|^1 = 309*16295346+38 by Lem1;
        then 2|^(19*4+1),38 are_congruent_mod 309 &
          38*38*38*38*2|^1 = 309*13496+8 by LemS;
        then 2|^(77*4+1),8 are_congruent_mod 309 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 310;
        2|^4 = 310*0+16 & 16*16*16*16*2|^3 = 310*1691+78 by Lm4,Lm3;
        then 2|^(4*4+3),78 are_congruent_mod 310 &
          78*78*78*78*2|^1 = 310*238806+252 by Lem1;
        then 2|^(19*4+1),252 are_congruent_mod 310 &
          252*252*252*252*2|^2 = 310*52035587+94 by LemS,Lm2;
        then 2|^(77*4+2),94 are_congruent_mod 310 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 311;
        hence thesis by XPRIMES1:311;
      end;
      suppose A: n = 312;
        2|^4 = 312*0+16 & 16*16*16*16*2|^3 = 312*1680+128 by Lm4,Lm3;
        then 2|^(4*4+3),128 are_congruent_mod 312 &
          128*128*128*128*2|^2 = 312*3441480+64 by Lem1,Lm2;
        then 2|^(19*4+2),64 are_congruent_mod 312 &
          64*64*64*64*2|^0 = 312*53773+40 by LemS,Lm1;
        then 2|^(78*4+0),40 are_congruent_mod 312 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 313;
        hence thesis by XPRIMES1:313;
      end;
      suppose A: n = 314;
        2|^4 = 314*0+16 & 16*16*16*16*2|^3 = 314*1669+222 by Lm4,Lm3;
        then 2|^(4*4+3),222 are_congruent_mod 314 &
          222*222*222*222*2|^2 = 314*30941562+156 by Lem1,Lm2;
        then 2|^(19*4+2),156 are_congruent_mod 314 &
          156*156*156*156*2|^2 = 314*7544470+4 by LemS;
        then 2|^(78*4+2),4 are_congruent_mod 314 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 315;
        2|^4 = 315*0+16 & 16*16*16*16*2|^3 = 315*1664+128 by Lm4,Lm3;
        then 2|^(4*4+3),128 are_congruent_mod 315 &
          128*128*128*128*2|^2 = 315*3408704+64 by Lem1,Lm2;
        then 2|^(19*4+2),64 are_congruent_mod 315 &
          64*64*64*64*2|^3 = 315*426088+8 by LemS,Lm3;
        then 2|^(78*4+3),8 are_congruent_mod 315 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 316;
        2|^4 = 316*0+16 & 16*16*16*16*2|^3 = 316*1659+44 by Lm4,Lm3;
        then 2|^(4*4+3),44 are_congruent_mod 316 &
          44*44*44*44*2|^3 = 316*94888+160 by Lem1;
        then 2|^(19*4+3),160 are_congruent_mod 316 &
          160*160*160*160*2|^0 = 316*2073924+16 by LemS,Lm1;
        then 2|^(79*4+0),16 are_congruent_mod 316 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 317;
        hence thesis by XPRIMES1:317;
      end;
      suppose A: n = 318;
        2|^4 = 318*0+16 & 16*16*16*16*2|^3 = 318*1648+224 by Lm4,Lm3;
        then 2|^(4*4+3),224 are_congruent_mod 318 &
          224*224*224*224*2|^3 = 318*63336628+104 by Lem1;
        then 2|^(19*4+3),104 are_congruent_mod 318 &
          104*104*104*104*2|^2 = 318*1471520+64 by LemS,Lm2;
        then 2|^(79*4+2),64 are_congruent_mod 318 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 319;
        2|^4 = 319*0+16 & 16*16*16*16*2|^3 = 319*1643+171 by Lm4,Lm3;
        then 2|^(4*4+3),171 are_congruent_mod 319 &
          171*171*171*171*2|^3 = 319*21442911+39 by Lem1;
        then 2|^(19*4+3),39 are_congruent_mod 319 &
          39*39*39*39*2|^3 = 319*58017+105 by LemS;
        then 2|^(79*4+3),105 are_congruent_mod 319 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 320;
        2|^5 = 320*0+32 & 32*32*32*32*2|^0 = 320*3276+256 by Lm5,Lm1;
        then 2|^(5*4+0),256 are_congruent_mod 320 &
          256*256*256*256*2|^0 = 320*13421772+256 by Lem1;
        then 2|^(20*4+0),256 are_congruent_mod 320 &
          256*256*256*256*2|^0 = 320*13421772+256 by LemS;
        then 2|^(80*4+0),256 are_congruent_mod 320 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 321;
        2|^5 = 321*0+32 & 32*32*32*32*2|^0 = 321*3266+190 by Lm5,Lm1;
        then 2|^(5*4+0),190 are_congruent_mod 321 &
          190*190*190*190*2|^0 = 321*4059844+76 by Lem1;
        then 2|^(20*4+0),76 are_congruent_mod 321 &
          76*76*76*76*2|^1 = 321*207864+8 by LemS;
        then 2|^(80*4+1),8 are_congruent_mod 321 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 322;
        2|^5 = 322*0+32 & 32*32*32*32*2|^0 = 322*3256+144 by Lm5,Lm1;
        then 2|^(5*4+0),144 are_congruent_mod 322 &
          144*144*144*144*2|^0 = 322*1335346+284 by Lem1;
        then 2|^(20*4+0),284 are_congruent_mod 322 &
          284*284*284*284*2|^2 = 322*80812302+100 by LemS,Lm2;
        then 2|^(80*4+2),100 are_congruent_mod 322 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 323;
        2|^5 = 323*0+32 & 32*32*32*32*2|^0 = 323*3246+118 by Lm5,Lm1;
        then 2|^(5*4+0),118 are_congruent_mod 323 &
          118*118*118*118*2|^0 = 323*600240+256 by Lem1;
        then 2|^(20*4+0),256 are_congruent_mod 323 &
          256*256*256*256*2|^3 = 323*106376898+314 by LemS,Lm3;
        then 2|^(80*4+3),314 are_congruent_mod 323 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 324;
        2|^5 = 324*0+32 & 32*32*32*32*2|^0 = 324*3236+112 by Lm5,Lm1;
        then 2|^(5*4+0),112 are_congruent_mod 324 &
          112*112*112*112*2|^1 = 324*971308+80 by Lem1;
        then 2|^(20*4+1),80 are_congruent_mod 324 &
          80*80*80*80*2|^0 = 324*126419+244 by LemS,Lm1;
        then 2|^(81*4+0),244 are_congruent_mod 324 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 325;
        2|^5 = 325*0+32 & 32*32*32*32*2|^0 = 325*3226+126 by Lm5,Lm1;
        then 2|^(5*4+0),126 are_congruent_mod 325 &
          126*126*126*126*2|^1 = 325*1551060+252 by Lem1;
        then 2|^(20*4+1),252 are_congruent_mod 325 &
          252*252*252*252*2|^1 = 325*24816972+132 by LemS;
        then 2|^(81*4+1),132 are_congruent_mod 325 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 326;
        2|^5 = 326*0+32 & 32*32*32*32*2|^0 = 326*3216+160 by Lm5,Lm1;
        then 2|^(5*4+0),160 are_congruent_mod 326 &
          160*160*160*160*2|^1 = 326*4020613+162 by Lem1;
        then 2|^(20*4+1),162 are_congruent_mod 326 &
          162*162*162*162*2|^2 = 326*8450890+4 by LemS,Lm2;
        then 2|^(81*4+2),4 are_congruent_mod 326 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 327;
        2|^5 = 327*0+32 & 32*32*32*32*2|^0 = 327*3206+214 by Lm5,Lm1;
        then 2|^(5*4+0),214 are_congruent_mod 327 &
          214*214*214*214*2|^1 = 327*12827361+185 by Lem1;
        then 2|^(20*4+1),185 are_congruent_mod 327 &
          185*185*185*185*2|^3 = 327*28656896+8 by LemS,Lm3;
        then 2|^(81*4+3),8 are_congruent_mod 327 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 328;
        2|^5 = 328*0+32 & 32*32*32*32*2|^0 = 328*3196+288 by Lm5,Lm1;
        then 2|^(5*4+0),288 are_congruent_mod 328 &
          288*288*288*288*2|^2 = 328*83898867+168 by Lem1,Lm2;
        then 2|^(20*4+2),168 are_congruent_mod 328 &
          168*168*168*168*2|^0 = 328*2428640+256 by LemS,Lm1;
        then 2|^(82*4+0),256 are_congruent_mod 328 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 329;
        2|^5 = 329*0+32 & 32*32*32*32*2|^0 = 329*3187+53 by Lm5,Lm1;
        then 2|^(5*4+0),53 are_congruent_mod 329 &
          53*53*53*53*2|^2 = 329*95932+296 by Lem1,Lm2;
        then 2|^(20*4+2),296 are_congruent_mod 329 &
          296*296*296*296*2|^1 = 329*46666039+81 by LemS;
        then 2|^(82*4+1),81 are_congruent_mod 329 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 330;
        2|^5 = 330*0+32 & 32*32*32*32*2|^0 = 330*3177+166 by Lm5,Lm1;
        then 2|^(5*4+0),166 are_congruent_mod 330 &
          166*166*166*166*2|^2 = 330*9204038+4 by Lem1,Lm2;
        then 2|^(20*4+2),4 are_congruent_mod 330 &
          4*4*4*4*2|^2 = 330*3+34 by LemS;
        then 2|^(82*4+2),34 are_congruent_mod 330 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 331;
        hence thesis by XPRIMES1:331;
      end;
      suppose A: n = 332;
        2|^5 = 332*0+32 & 32*32*32*32*2|^0 = 332*3158+120 by Lm5,Lm1;
        then 2|^(5*4+0),120 are_congruent_mod 332 &
          120*120*120*120*2|^3 = 332*4996626+168 by Lem1,Lm3;
        then 2|^(20*4+3),168 are_congruent_mod 332 &
          168*168*168*168*2|^0 = 332*2399380+16 by LemS,Lm1;
        then 2|^(83*4+0),16 are_congruent_mod 332 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 333;
        2|^5 = 333*0+32 & 32*32*32*32*2|^0 = 333*3148+292 by Lm5,Lm1;
        then 2|^(5*4+0),292 are_congruent_mod 333 &
          292*292*292*292*2|^3 = 333*174653446+50 by Lem1,Lm3;
        then 2|^(20*4+3),50 are_congruent_mod 333 &
          50*50*50*50*2|^1 = 333*37537+179 by LemS;
        then 2|^(83*4+1),179 are_congruent_mod 333 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 334;
        2|^5 = 334*0+32 & 32*32*32*32*2|^0 = 334*3139+150 by Lm5,Lm1;
        then 2|^(5*4+0),150 are_congruent_mod 334 &
          150*150*150*150*2|^3 = 334*12125748+168 by Lem1,Lm3;
        then 2|^(20*4+3),168 are_congruent_mod 334 &
          168*168*168*168*2|^2 = 334*9540050+4 by LemS,Lm2;
        then 2|^(83*4+2),4 are_congruent_mod 334 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 335;
        2|^5 = 335*0+32 & 32*32*32*32*2|^0 = 335*3130+26 by Lm5,Lm1;
        then 2|^(5*4+0),26 are_congruent_mod 335 &
          26*26*26*26*2|^3 = 335*10912+288 by Lem1,Lm3;
        then 2|^(20*4+3),288 are_congruent_mod 335 &
          288*288*288*288*2|^3 = 335*164291513+233 by LemS;
        then 2|^(83*4+3),233 are_congruent_mod 335 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 336;
        2|^5 = 336*0+32 & 32*32*32*32*2|^1 = 336*6241+176 by Lm5;
        then 2|^(5*4+1),176 are_congruent_mod 336 &
          176*176*176*176*2|^0 = 336*2855692+64 by Lem1,Lm1;
        then 2|^(21*4+0),64 are_congruent_mod 336 &
          64*64*64*64*2|^0 = 336*49932+64 by LemS;
        then 2|^(84*4+0),64 are_congruent_mod 336 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 337;
        hence thesis by XPRIMES1:337;
      end;
      suppose A: n = 338;
        2|^5 = 338*0+32 & 32*32*32*32*2|^1 = 338*6204+200 by Lm5;
        then 2|^(5*4+1),200 are_congruent_mod 338 &
          200*200*200*200*2|^0 = 338*4733727+274 by Lem1,Lm1;
        then 2|^(21*4+0),274 are_congruent_mod 338 &
          274*274*274*274*2|^2 = 338*66703026+316 by LemS,Lm2;
        then 2|^(84*4+2),316 are_congruent_mod 338 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 339;
        2|^5 = 339*0+32 & 32*32*32*32*2|^1 = 339*6186+98 by Lm5;
        then 2|^(5*4+1),98 are_congruent_mod 339 &
          98*98*98*98*2|^0 = 339*272085+1 by Lem1,Lm1;
        then 2|^(21*4+0),1 are_congruent_mod 339 &
          1*1*1*1*2|^3 = 339*0+8 by LemS,Lm3;
        then 2|^(84*4+3),8 are_congruent_mod 339 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 340;
        2|^5 = 340*0+32 & 32*32*32*32*2|^1 = 340*6168+32 by Lm5;
        then 2|^(5*4+1),32 are_congruent_mod 340 &
          32*32*32*32*2|^1 = 340*6168+32 by Lem1;
        then 2|^(21*4+1),32 are_congruent_mod 340 &
          32*32*32*32*2|^0 = 340*3084+16 by LemS,Lm1;
        then 2|^(85*4+0),16 are_congruent_mod 340 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 341;
        hence thesis by A5,ENUMSET1:def 2;
      end;
      suppose A: n = 342;
        2|^5 = 342*0+32 & 32*32*32*32*2|^1 = 342*6132+8 by Lm5;
        then 2|^(5*4+1),8 are_congruent_mod 342 &
          8*8*8*8*2|^1 = 342*23+326 by Lem1;
        then 2|^(21*4+1),326 are_congruent_mod 342 &
          326*326*326*326*2|^2 = 342*132100446+172 by LemS,Lm2;
        then 2|^(85*4+2),172 are_congruent_mod 342 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 343;
        2|^5 = 343*0+32 & 32*32*32*32*2|^1 = 343*6114+50 by Lm5;
        then 2|^(5*4+1),50 are_congruent_mod 343 &
          50*50*50*50*2|^1 = 343*36443+51 by Lem1;
        then 2|^(21*4+1),51 are_congruent_mod 343 &
          51*51*51*51*2|^3 = 343*157788+324 by LemS,Lm3;
        then 2|^(85*4+3),324 are_congruent_mod 343 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 344;
        2|^5 = 344*0+32 & 32*32*32*32*2|^1 = 344*6096+128 by Lm5;
        then 2|^(5*4+1),128 are_congruent_mod 344 &
          128*128*128*128*2|^2 = 344*3121342+176 by Lem1,Lm2;
        then 2|^(21*4+2),176 are_congruent_mod 344 &
          176*176*176*176*2|^0 = 344*2789280+256 by LemS,Lm1;
        then 2|^(86*4+0),256 are_congruent_mod 344 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 345;
        2|^5 = 345*0+32 & 32*32*32*32*2|^1 = 345*6078+242 by Lm5;
        then 2|^(5*4+1),242 are_congruent_mod 345 &
          242*242*242*242*2|^2 = 345*39765125+259 by Lem1,Lm2;
        then 2|^(21*4+2),259 are_congruent_mod 345 &
          259*259*259*259*2|^1 = 345*26086148+62 by LemS;
        then 2|^(86*4+1),62 are_congruent_mod 345 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 346;
        2|^5 = 346*0+32 & 32*32*32*32*2|^1 = 346*6061+46 by Lm5;
        then 2|^(5*4+1),46 are_congruent_mod 346 &
          46*46*46*46*2|^2 = 346*51762+172 by Lem1,Lm2;
        then 2|^(21*4+2),172 are_congruent_mod 346 &
          172*172*172*172*2|^2 = 346*10118070+4 by LemS;
        then 2|^(86*4+2),4 are_congruent_mod 346 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 347;
        hence thesis by XPRIMES1:347;
      end;
      suppose A: n = 348;
        2|^5 = 348*0+32 & 32*32*32*32*2|^1 = 348*6026+104 by Lm5;
        then 2|^(5*4+1),104 are_congruent_mod 348 &
          104*104*104*104*2|^3 = 348*2689330+8 by Lem1,Lm3;
        then 2|^(21*4+3),8 are_congruent_mod 348 &
          8*8*8*8*2|^0 = 348*11+268 by LemS,Lm1;
        then 2|^(87*4+0),268 are_congruent_mod 348 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 349;
        hence thesis by XPRIMES1:349;
      end;
      suppose A: n = 350;
        2|^5 = 350*0+32 & 32*32*32*32*2|^1 = 350*5991+302 by Lm5;
        then 2|^(5*4+1),302 are_congruent_mod 350 &
          302*302*302*302*2|^3 = 350*190129591+78 by Lem1,Lm3;
        then 2|^(21*4+3),78 are_congruent_mod 350 &
          78*78*78*78*2|^2 = 350*423029+74 by LemS,Lm2;
        then 2|^(87*4+2),74 are_congruent_mod 350 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 351;
        2|^5 = 351*0+32 & 32*32*32*32*2|^1 = 351*5974+278 by Lm5;
        then 2|^(5*4+1),278 are_congruent_mod 351 &
          278*278*278*278*2|^3 = 351*136132573+125 by Lem1,Lm3;
        then 2|^(21*4+3),125 are_congruent_mod 351 &
          125*125*125*125*2|^3 = 351*5564458+242 by LemS;
        then 2|^(87*4+3),242 are_congruent_mod 351 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 352;
        2|^5 = 352*0+32 & 32*32*32*32*2|^2 = 352*11915+224 by Lm5,Lm2;
        then 2|^(5*4+2),224 are_congruent_mod 352 &
          224*224*224*224*2|^0 = 352*7152360+256 by Lem1,Lm1;
        then 2|^(22*4+0),256 are_congruent_mod 352 &
          256*256*256*256*2|^0 = 352*12201611+224 by LemS;
        then 2|^(88*4+0),224 are_congruent_mod 352 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 353;
        hence thesis by XPRIMES1:353;
      end;
      suppose A: n = 354;
        2|^5 = 354*0+32 & 32*32*32*32*2|^2 = 354*11848+112 by Lm5,Lm2;
        then 2|^(5*4+2),112 are_congruent_mod 354 &
          112*112*112*112*2|^0 = 354*444496+352 by Lem1,Lm1;
        then 2|^(22*4+0),352 are_congruent_mod 354 &
          352*352*352*352*2|^2 = 354*173471200+64 by LemS,Lm2;
        then 2|^(88*4+2),64 are_congruent_mod 354 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 355;
        2|^5 = 355*0+32 & 32*32*32*32*2|^2 = 355*11814+334 by Lm5,Lm2;
        then 2|^(5*4+2),334 are_congruent_mod 355 &
          334*334*334*334*2|^0 = 355*35055608+296 by Lem1,Lm1;
        then 2|^(22*4+0),296 are_congruent_mod 355 &
          296*296*296*296*2|^3 = 355*172992979+103 by LemS,Lm3;
        then 2|^(88*4+3),103 are_congruent_mod 355 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 356;
        2|^5 = 356*0+32 & 32*32*32*32*2|^2 = 356*11781+268 by Lm5,Lm2;
        then 2|^(5*4+2),268 are_congruent_mod 356 &
          268*268*268*268*2|^1 = 356*28981387+180 by Lem1;
        then 2|^(22*4+1),180 are_congruent_mod 356 &
          180*180*180*180*2|^0 = 356*2948764+16 by LemS,Lm1;
        then 2|^(89*4+0),16 are_congruent_mod 356 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 357;
        2|^5 = 357*0+32 & 32*32*32*32*2|^2 = 357*11748+268 by Lm5,Lm2;
        then 2|^(5*4+2),268 are_congruent_mod 357 &
          268*268*268*268*2|^1 = 357*28900207+53 by Lem1;
        then 2|^(22*4+1),53 are_congruent_mod 357 &
          53*53*53*53*2|^1 = 357*44204+134 by LemS;
        then 2|^(89*4+1),134 are_congruent_mod 357 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 358;
        2|^5 = 358*0+32 & 32*32*32*32*2|^2 = 358*11715+334 by Lm5,Lm2;
        then 2|^(5*4+2),334 are_congruent_mod 358 &
          334*334*334*334*2|^1 = 358*69523693+178 by Lem1;
        then 2|^(22*4+1),178 are_congruent_mod 358 &
          178*178*178*178*2|^2 = 358*11216490+4 by LemS,Lm2;
        then 2|^(89*4+2),4 are_congruent_mod 358 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 359;
        hence thesis by XPRIMES1:359;
      end;
      suppose A: n = 360;
        2|^5 = 360*0+32 & 32*32*32*32*2|^2 = 360*11650+304 by Lm5,Lm2;
        then 2|^(5*4+2),304 are_congruent_mod 360 &
          304*304*304*304*2|^2 = 360*94896856+64 by Lem1;
        then 2|^(22*4+2),64 are_congruent_mod 360 &
          64*64*64*64*2|^0 = 360*46603+136 by LemS,Lm1;
        then 2|^(90*4+0),136 are_congruent_mod 360 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 361;
        2|^5 = 361*0+32 & 32*32*32*32*2|^2 = 361*11618+206 by Lm5,Lm2;
        then 2|^(5*4+2),206 are_congruent_mod 361 &
          206*206*206*206*2|^2 = 361*19953618+286 by Lem1;
        then 2|^(22*4+2),286 are_congruent_mod 361 &
          286*286*286*286*2|^1 = 361*37066956+116 by LemS;
        then 2|^(90*4+1),116 are_congruent_mod 361 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 362;
        2|^5 = 362*0+32 & 32*32*32*32*2|^2 = 362*11586+172 by Lm5,Lm2;
        then 2|^(5*4+2),172 are_congruent_mod 362 &
          172*172*172*172*2|^2 = 362*9670862+180 by Lem1;
        then 2|^(22*4+2),180 are_congruent_mod 362 &
          180*180*180*180*2|^2 = 362*11599558+4 by LemS;
        then 2|^(90*4+2),4 are_congruent_mod 362 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 363;
        2|^5 = 363*0+32 & 32*32*32*32*2|^2 = 363*11554+202 by Lm5,Lm2;
        then 2|^(5*4+2),202 are_congruent_mod 363 &
          202*202*202*202*2|^2 = 363*18346737+133 by Lem1;
        then 2|^(22*4+2),133 are_congruent_mod 363 &
          133*133*133*133*2|^3 = 363*6895883+239 by LemS,Lm3;
        then 2|^(90*4+3),239 are_congruent_mod 363 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 364;
        2|^5 = 364*0+32 & 32*32*32*32*2|^2 = 364*11522+296 by Lm5,Lm2;
        then 2|^(5*4+2),296 are_congruent_mod 364 &
          296*296*296*296*2|^3 = 364*168715680+128 by Lem1,Lm3;
        then 2|^(22*4+3),128 are_congruent_mod 364 &
          128*128*128*128*2|^0 = 364*737460+16 by LemS,Lm1;
        then 2|^(91*4+0),16 are_congruent_mod 364 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 365;
        2|^5 = 365*0+32 & 32*32*32*32*2|^2 = 365*11491+89 by Lm5,Lm2;
        then 2|^(5*4+2),89 are_congruent_mod 365 &
          89*89*89*89*2|^3 = 365*1375172+148 by Lem1,Lm3;
        then 2|^(22*4+3),148 are_congruent_mod 365 &
          148*148*148*148*2|^1 = 365*2628960+32 by LemS;
        then 2|^(91*4+1),32 are_congruent_mod 365 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 366;
        2|^5 = 366*0+32 & 32*32*32*32*2|^2 = 366*11459+310 by Lm5,Lm2;
        then 2|^(5*4+2),310 are_congruent_mod 366 &
          310*310*310*310*2|^3 = 366*201862513+242 by Lem1,Lm3;
        then 2|^(22*4+3),242 are_congruent_mod 366 &
          242*242*242*242*2|^2 = 366*37483520+64 by LemS,Lm2;
        then 2|^(91*4+2),64 are_congruent_mod 366 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 367;
        hence thesis by XPRIMES1:367;
      end;
      suppose A: n = 368;
        2|^5 = 368*0+32 & 32*32*32*32*2|^3 = 368*22795+48 by Lm5,Lm3;
        then 2|^(5*4+3),48 are_congruent_mod 368 &
          48*48*48*48*2|^0 = 368*14425+16 by Lem1,Lm1;
        then 2|^(23*4+0),16 are_congruent_mod 368 &
          16*16*16*16*2|^0 = 368*178+32 by LemS;
        then 2|^(92*4+0),32 are_congruent_mod 368 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 369;
        2|^5 = 369*0+32 & 32*32*32*32*2|^3 = 369*22733+131 by Lm5,Lm3;
        then 2|^(5*4+3),131 are_congruent_mod 369 &
          131*131*131*131*2|^0 = 369*798102+283 by Lem1,Lm1;
        then 2|^(23*4+0),283 are_congruent_mod 369 &
          283*283*283*283*2|^1 = 369*34765571+143 by LemS;
        then 2|^(92*4+1),143 are_congruent_mod 369 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 370;
        2|^5 = 370*0+32 & 32*32*32*32*2|^3 = 370*22671+338 by Lm5,Lm3;
        then 2|^(5*4+3),338 are_congruent_mod 370 &
          338*338*338*338*2|^0 = 370*35274841+366 by Lem1,Lm1;
        then 2|^(23*4+0),366 are_congruent_mod 370 &
          366*366*366*366*2|^2 = 370*193991458+284 by LemS,Lm2;
        then 2|^(92*4+2),284 are_congruent_mod 370 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 371;
        2|^5 = 371*0+32 & 32*32*32*32*2|^3 = 371*22610+298 by Lm5,Lm3;
        then 2|^(5*4+3),298 are_congruent_mod 371 &
          298*298*298*298*2|^0 = 371*21256470+46 by Lem1,Lm1;
        then 2|^(23*4+0),46 are_congruent_mod 371 &
          46*46*46*46*2|^3 = 371*96548+340 by LemS,Lm3;
        then 2|^(92*4+3),340 are_congruent_mod 371 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 372;
        2|^5 = 372*0+32 & 32*32*32*32*2|^3 = 372*22550+8 by Lm5,Lm3;
        then 2|^(5*4+3),8 are_congruent_mod 372 &
          8*8*8*8*2|^1 = 372*22+8 by Lem1;
        then 2|^(23*4+1),8 are_congruent_mod 372 &
          8*8*8*8*2|^0 = 372*11+4 by LemS,Lm1;
        then 2|^(93*4+0),4 are_congruent_mod 372 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 373;
        hence thesis by XPRIMES1:373;
      end;
      suppose A: n = 374;
        2|^5 = 374*0+32 & 32*32*32*32*2|^3 = 374*22429+162 by Lm5,Lm3;
        then 2|^(5*4+3),162 are_congruent_mod 374 &
          162*162*162*162*2|^1 = 374*3683141+338 by Lem1;
        then 2|^(23*4+1),338 are_congruent_mod 374 &
          338*338*338*338*2|^2 = 374*139590283+302 by LemS,Lm2;
        then 2|^(93*4+2),302 are_congruent_mod 374 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 375;
        2|^5 = 375*0+32 & 32*32*32*32*2|^3 = 375*22369+233 by Lm5,Lm3;
        then 2|^(5*4+3),233 are_congruent_mod 375 &
          233*233*233*233*2|^1 = 375*15718909+167 by Lem1;
        then 2|^(23*4+1),167 are_congruent_mod 375 &
          167*167*167*167*2|^3 = 375*16592988+68 by LemS,Lm3;
        then 2|^(93*4+3),68 are_congruent_mod 375 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 376;
        2|^5 = 376*0+32 & 32*32*32*32*2|^3 = 376*22310+48 by Lm5,Lm3;
        then 2|^(5*4+3),48 are_congruent_mod 376 &
          48*48*48*48*2|^2 = 376*56472+192 by Lem1,Lm2;
        then 2|^(23*4+2),192 are_congruent_mod 376 &
          192*192*192*192*2|^0 = 376*3614240+256 by LemS,Lm1;
        then 2|^(94*4+0),256 are_congruent_mod 376 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 377;
        2|^5 = 377*0+32 & 32*32*32*32*2|^3 = 377*22250+358 by Lm5,Lm3;
        then 2|^(5*4+3),358 are_congruent_mod 377 &
          358*358*358*358*2|^2 = 377*174281282+270 by Lem1,Lm2;
        then 2|^(23*4+2),270 are_congruent_mod 377 &
          270*270*270*270*2|^1 = 377*28193156+188 by LemS;
        then 2|^(94*4+1),188 are_congruent_mod 377 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 378;
        2|^5 = 378*0+32 & 32*32*32*32*2|^3 = 378*22192+32 by Lm5,Lm3;
        then 2|^(5*4+3),32 are_congruent_mod 378 &
          32*32*32*32*2|^2 = 378*11096+16 by Lem1,Lm2;
        then 2|^(23*4+2),16 are_congruent_mod 378 &
          16*16*16*16*2|^2 = 378*693+190 by LemS;
        then 2|^(94*4+2),190 are_congruent_mod 378 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 379;
        hence thesis by XPRIMES1:379;
      end;
      suppose A: n = 380;
        2|^5 = 380*0+32 & 32*32*32*32*2|^3 = 380*22075+108 by Lm5,Lm3;
        then 2|^(5*4+3),108 are_congruent_mod 380 &
          108*108*108*108*2|^3 = 380*2864187+108 by Lem1;
        then 2|^(23*4+3),108 are_congruent_mod 380 &
          108*108*108*108*2|^0 = 380*358023+156 by LemS,Lm1;
        then 2|^(95*4+0),156 are_congruent_mod 380 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 381;
        2|^5 = 381*0+32 & 32*32*32*32*2|^3 = 381*22017+131 by Lm5,Lm3;
        then 2|^(5*4+3),131 are_congruent_mod 381 &
          131*131*131*131*2|^3 = 381*6183725+143 by Lem1;
        then 2|^(23*4+3),143 are_congruent_mod 381 &
          143*143*143*143*2|^1 = 381*2195074+8 by LemS;
        then 2|^(95*4+1),8 are_congruent_mod 381 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 382;
        2|^5 = 382*0+32 & 32*32*32*32*2|^3 = 382*21959+270 by Lm5,Lm3;
        then 2|^(5*4+3),270 are_congruent_mod 382 &
          270*270*270*270*2|^3 = 382*111296544+192 by Lem1;
        then 2|^(23*4+3),192 are_congruent_mod 382 &
          192*192*192*192*2|^2 = 382*14229890+4 by LemS,Lm2;
        then 2|^(95*4+2),4 are_congruent_mod 382 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 383;
        hence thesis by XPRIMES1:383;
      end;
      suppose A: n = 384;
        2|^6 = 384*0+64 & 64*64*64*64*2|^0 = 384*43690+256 by Lm6,Lm1;
        then 2|^(6*4+0),256 are_congruent_mod 384 &
          256*256*256*256*2|^0 = 384*11184810+256 by Lem1;
        then 2|^(24*4+0),256 are_congruent_mod 384 &
          256*256*256*256*2|^0 = 384*11184810+256 by LemS;
        then 2|^(96*4+0),256 are_congruent_mod 384 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 385;
        2|^6 = 385*0+64 & 64*64*64*64*2|^0 = 385*43577+71 by Lm6,Lm1;
        then 2|^(6*4+0),71 are_congruent_mod 385 &
          71*71*71*71*2|^0 = 385*66004+141 by Lem1;
        then 2|^(24*4+0),141 are_congruent_mod 385 &
          141*141*141*141*2|^1 = 385*2053268+142 by LemS;
        then 2|^(96*4+1),142 are_congruent_mod 385 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 386;
        2|^6 = 386*0+64 & 64*64*64*64*2|^0 = 386*43464+112 by Lm6,Lm1;
        then 2|^(6*4+0),112 are_congruent_mod 386 &
          112*112*112*112*2|^0 = 386*407647+194 by Lem1;
        then 2|^(24*4+0),194 are_congruent_mod 386 &
          194*194*194*194*2|^2 = 386*14678430+4 by LemS,Lm2;
        then 2|^(96*4+2),4 are_congruent_mod 386 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 387;
        2|^6 = 387*0+64 & 64*64*64*64*2|^0 = 387*43351+379 by Lm6,Lm1;
        then 2|^(6*4+0),379 are_congruent_mod 387 &
          379*379*379*379*2|^0 = 387*53314565+226 by Lem1;
        then 2|^(24*4+0),226 are_congruent_mod 387 &
          226*226*226*226*2|^3 = 387*53927809+125 by LemS,Lm3;
        then 2|^(96*4+3),125 are_congruent_mod 387 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 388;
        2|^6 = 388*0+64 & 64*64*64*64*2|^0 = 388*43240+96 by Lm6,Lm1;
        then 2|^(6*4+0),96 are_congruent_mod 388 &
          96*96*96*96*2|^1 = 388*437807+196 by Lem1;
        then 2|^(24*4+1),196 are_congruent_mod 388 &
          196*196*196*196*2|^0 = 388*3803580+16 by LemS,Lm1;
        then 2|^(97*4+0),16 are_congruent_mod 388 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 389;
        hence thesis by XPRIMES1:389;
      end;
      suppose A: n = 390;
        2|^6 = 390*0+64 & 64*64*64*64*2|^0 = 390*43018+196 by Lm6,Lm1;
        then 2|^(6*4+0),196 are_congruent_mod 390 &
          196*196*196*196*2|^1 = 390*7568149+2 by Lem1;
        then 2|^(24*4+1),2 are_congruent_mod 390 &
          2*2*2*2*2|^2 = 390*0+64 by LemS,Lm2;
        then 2|^(97*4+2),64 are_congruent_mod 390 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 391;
        2|^6 = 391*0+64 & 64*64*64*64*2|^0 = 391*42908+188 by Lm6,Lm1;
        then 2|^(6*4+0),188 are_congruent_mod 391 &
          188*188*188*188*2|^1 = 391*6389761+121 by Lem1;
        then 2|^(24*4+1),121 are_congruent_mod 391 &
          121*121*121*121*2|^3 = 391*4385859+179 by LemS,Lm3;
        then 2|^(97*4+3),179 are_congruent_mod 391 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 392;
        2|^6 = 392*0+64 & 64*64*64*64*2|^0 = 392*42799+8 by Lm6,Lm1;
        then 2|^(6*4+0),8 are_congruent_mod 392 &
          8*8*8*8*2|^2 = 392*41+312 by Lem1,Lm2;
        then 2|^(24*4+2),312 are_congruent_mod 392 &
          312*312*312*312*2|^0 = 392*24173097+312 by LemS,Lm1;
        then 2|^(98*4+0),312 are_congruent_mod 392 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 393;
        2|^6 = 393*0+64 & 64*64*64*64*2|^0 = 393*42690+46 by Lm6,Lm1;
        then 2|^(6*4+0),46 are_congruent_mod 393 &
          46*46*46*46*2|^2 = 393*45572+28 by Lem1,Lm2;
        then 2|^(24*4+2),28 are_congruent_mod 393 &
          28*28*28*28*2|^1 = 393*3128+8 by LemS;
        then 2|^(98*4+1),8 are_congruent_mod 393 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 394;
        2|^6 = 394*0+64 & 64*64*64*64*2|^0 = 394*42581+302 by Lm6,Lm1;
        then 2|^(6*4+0),302 are_congruent_mod 394 &
          302*302*302*302*2|^2 = 394*84448422+196 by Lem1,Lm2;
        then 2|^(24*4+2),196 are_congruent_mod 394 &
          196*196*196*196*2|^2 = 394*14982630+4 by LemS;
        then 2|^(98*4+2),4 are_congruent_mod 394 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 395;
        2|^6 = 395*0+64 & 64*64*64*64*2|^0 = 395*42473+381 by Lm6,Lm1;
        then 2|^(6*4+0),381 are_congruent_mod 395 &
          381*381*381*381*2|^2 = 395*213384465+9 by Lem1,Lm2;
        then 2|^(24*4+2),9 are_congruent_mod 395 &
          9*9*9*9*2|^3 = 395*132+348 by LemS,Lm3;
        then 2|^(98*4+3),348 are_congruent_mod 395 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 396;
        2|^6 = 396*0+64 & 64*64*64*64*2|^0 = 396*42366+280 by Lm6,Lm1;
        then 2|^(6*4+0),280 are_congruent_mod 396 &
          280*280*280*280*2|^3 = 396*124172929+116 by Lem1,Lm3;
        then 2|^(24*4+3),116 are_congruent_mod 396 &
          116*116*116*116*2|^0 = 396*457232+64 by LemS,Lm1;
        then 2|^(99*4+0),64 are_congruent_mod 396 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 397;
        hence thesis by XPRIMES1:397;
      end;
      suppose A: n = 398;
        2|^6 = 398*0+64 & 64*64*64*64*2|^0 = 398*42153+322 by Lm6,Lm1;
        then 2|^(6*4+0),322 are_congruent_mod 398 &
          322*322*322*322*2|^3 = 398*216087876+200 by Lem1,Lm3;
        then 2|^(24*4+3),200 are_congruent_mod 398 &
          200*200*200*200*2|^2 = 398*16080402+4 by LemS,Lm2;
        then 2|^(99*4+2),4 are_congruent_mod 398 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 399;
        2|^6 = 399*0+64 & 64*64*64*64*2|^0 = 399*42048+64 by Lm6,Lm1;
        then 2|^(6*4+0),64 are_congruent_mod 399 &
          64*64*64*64*2|^3 = 399*336385+113 by Lem1,Lm3;
        then 2|^(24*4+3),113 are_congruent_mod 399 &
          113*113*113*113*2|^3 = 399*3269120+8 by LemS;
        then 2|^(99*4+3),8 are_congruent_mod 399 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 400;
        2|^6 = 400*0+64 & 64*64*64*64*2|^1 = 400*83886+32 by Lm6;
        then 2|^(6*4+1),32 are_congruent_mod 400 &
          32*32*32*32*2|^0 = 400*2621+176 by Lem1,Lm1;
        then 2|^(25*4+0),176 are_congruent_mod 400 &
          176*176*176*176*2|^0 = 400*2398781+176 by LemS;
        then 2|^(100*4+0),176 are_congruent_mod 400 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 401;
        hence thesis by XPRIMES1:401;
      end;
      suppose A: n = 402;
        2|^6 = 402*0+64 & 64*64*64*64*2|^1 = 402*83468+296 by Lm6;
        then 2|^(6*4+1),296 are_congruent_mod 402 &
          296*296*296*296*2|^0 = 402*19095928+400 by Lem1,Lm1;
        then 2|^(25*4+0),400 are_congruent_mod 402 &
          400*400*400*400*2|^2 = 402*254726368+64 by LemS,Lm2;
        then 2|^(100*4+2),64 are_congruent_mod 402 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 403;
        2|^6 = 403*0+64 & 64*64*64*64*2|^1 = 403*83261+249 by Lm6;
        then 2|^(6*4+1),249 are_congruent_mod 403 &
          249*249*249*249*2|^0 = 403*9538769+94 by Lem1,Lm1;
        then 2|^(25*4+0),94 are_congruent_mod 403 &
          94*94*94*94*2|^3 = 403*1549873+349 by LemS,Lm3;
        then 2|^(100*4+3),349 are_congruent_mod 403 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 404;
        2|^6 = 404*0+64 & 64*64*64*64*2|^1 = 404*83055+212 by Lm6;
        then 2|^(6*4+1),212 are_congruent_mod 404 &
          212*212*212*212*2|^1 = 404*9999817+204 by Lem1;
        then 2|^(25*4+1),204 are_congruent_mod 404 &
          204*204*204*204*2|^0 = 404*4286860+16 by LemS,Lm1;
        then 2|^(101*4+0),16 are_congruent_mod 404 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 404 < n & n <= 504;
      then 404+1 <= n & n <= 504 by NAT_1:13;
      then n = 405 or ... or n = 504;
      then per cases;
      suppose A: n = 405;
        2|^6 = 405*0+64 & 64*64*64*64*2|^1 = 405*82850+182 by Lm6;
        then 2|^(6*4+1),182 are_congruent_mod 405 &
          182*182*182*182*2|^1 = 405*5418268+212 by Lem1;
        then 2|^(25*4+1),212 are_congruent_mod 405 &
          212*212*212*212*2|^1 = 405*9975126+242 by LemS;
        then 2|^(101*4+1),242 are_congruent_mod 405 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 406;
        2|^6 = 406*0+64 & 64*64*64*64*2|^1 = 406*82646+156 by Lm6;
        then 2|^(6*4+1),156 are_congruent_mod 406 &
          156*156*156*156*2|^1 = 406*2917442+340 by Lem1;
        then 2|^(25*4+1),340 are_congruent_mod 406 &
          340*340*340*340*2|^2 = 406*131658719+86 by LemS,Lm2;
        then 2|^(101*4+2),86 are_congruent_mod 406 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 407;
        2|^6 = 407*0+64 & 64*64*64*64*2|^1 = 407*82443+131 by Lm6;
        then 2|^(6*4+1),131 are_congruent_mod 407 &
          131*131*131*131*2|^1 = 407*1447174+24 by Lem1;
        then 2|^(25*4+1),24 are_congruent_mod 407 &
          24*24*24*24*2|^3 = 407*6521+161 by LemS,Lm3;
        then 2|^(101*4+3),161 are_congruent_mod 407 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 408;
        2|^6 = 408*0+64 & 64*64*64*64*2|^1 = 408*82241+104 by Lm6;
        then 2|^(6*4+1),104 are_congruent_mod 408 &
          104*104*104*104*2|^2 = 408*1146920+64 by Lem1,Lm2;
        then 2|^(25*4+2),64 are_congruent_mod 408 &
          64*64*64*64*2|^0 = 408*41120+256 by LemS,Lm1;
        then 2|^(102*4+0),256 are_congruent_mod 408 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 409;
        hence thesis by XPRIMES1:409;
      end;
      suppose A: n = 410;
        2|^6 = 410*0+64 & 64*64*64*64*2|^1 = 410*81840+32 by Lm6;
        then 2|^(6*4+1),32 are_congruent_mod 410 &
          32*32*32*32*2|^2 = 410*10230+4 by Lem1,Lm2;
        then 2|^(25*4+2),4 are_congruent_mod 410 &
          4*4*4*4*2|^2 = 410*2+204 by LemS;
        then 2|^(102*4+2),204 are_congruent_mod 410 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 411;
        2|^6 = 411*0+64 & 64*64*64*64*2|^1 = 411*81640+392 by Lm6;
        then 2|^(6*4+1),392 are_congruent_mod 411 &
          392*392*392*392*2|^2 = 411*229806568+136 by Lem1,Lm2;
        then 2|^(25*4+2),136 are_congruent_mod 411 &
          136*136*136*136*2|^3 = 411*6658920+8 by LemS,Lm3;
        then 2|^(102*4+3),8 are_congruent_mod 411 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 412;
        2|^6 = 412*0+64 & 64*64*64*64*2|^1 = 412*81442+328 by Lm6;
        then 2|^(6*4+1),328 are_congruent_mod 412 &
          328*328*328*328*2|^3 = 412*224744020+208 by Lem1,Lm3;
        then 2|^(25*4+3),208 are_congruent_mod 412 &
          208*208*208*208*2|^0 = 412*4543140+16 by LemS,Lm1;
        then 2|^(103*4+0),16 are_congruent_mod 412 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 413;
        2|^6 = 413*0+64 & 64*64*64*64*2|^1 = 413*81245+247 by Lm6;
        then 2|^(6*4+1),247 are_congruent_mod 413 &
          247*247*247*247*2|^3 = 413*72098752+72 by Lem1,Lm3;
        then 2|^(25*4+3),72 are_congruent_mod 413 &
          72*72*72*72*2|^1 = 413*130139+305 by LemS;
        then 2|^(103*4+1),305 are_congruent_mod 413 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 414;
        2|^6 = 414*0+64 & 64*64*64*64*2|^1 = 414*81049+146 by Lm6;
        then 2|^(6*4+1),146 are_congruent_mod 414 &
          146*146*146*146*2|^3 = 414*8780132+200 by Lem1,Lm3;
        then 2|^(25*4+3),200 are_congruent_mod 414 &
          200*200*200*200*2|^2 = 414*15458937+82 by LemS,Lm2;
        then 2|^(103*4+2),82 are_congruent_mod 414 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 415;
        2|^6 = 415*0+64 & 64*64*64*64*2|^1 = 415*80854+22 by Lm6;
        then 2|^(6*4+1),22 are_congruent_mod 415 &
          22*22*22*22*2|^3 = 415*4515+323 by Lem1,Lm3;
        then 2|^(25*4+3),323 are_congruent_mod 415 &
          323*323*323*323*2|^3 = 415*209822462+198 by LemS;
        then 2|^(103*4+3),198 are_congruent_mod 415 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 416;
        2|^6 = 416*0+64 & 64*64*64*64*2|^2 = 416*161319+160 by Lm6,Lm2;
        then 2|^(6*4+2),160 are_congruent_mod 416 &
          160*160*160*160*2|^0 = 416*1575384+256 by Lem1,Lm1;
        then 2|^(26*4+0),256 are_congruent_mod 416 &
          256*256*256*256*2|^0 = 416*10324440+256 by LemS;
        then 2|^(104*4+0),256 are_congruent_mod 416 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 417;
        2|^6 = 417*0+64 & 64*64*64*64*2|^2 = 417*160932+220 by Lm6,Lm2;
        then 2|^(6*4+2),220 are_congruent_mod 417 &
          220*220*220*220*2|^0 = 417*5617649+367 by Lem1,Lm1;
        then 2|^(26*4+0),367 are_congruent_mod 417 &
          367*367*367*367*2|^1 = 417*87007802+8 by LemS;
        then 2|^(104*4+1),8 are_congruent_mod 417 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 418;
        2|^6 = 418*0+64 & 64*64*64*64*2|^2 = 418*160547+218 by Lm6,Lm2;
        then 2|^(6*4+2),218 are_congruent_mod 418 &
          218*218*218*218*2|^0 = 418*5403183+82 by Lem1,Lm1;
        then 2|^(26*4+0),82 are_congruent_mod 418 &
          82*82*82*82*2|^2 = 418*432652+168 by LemS,Lm2;
        then 2|^(104*4+2),168 are_congruent_mod 418 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 419;
        hence thesis by XPRIMES1:419;
      end;
      suppose A: n = 420;
        2|^6 = 420*0+64 & 64*64*64*64*2|^2 = 420*159783+4 by Lm6,Lm2;
        then 2|^(6*4+2),4 are_congruent_mod 420 &
          4*4*4*4*2|^1 = 420*1+92 by Lem1;
        then 2|^(26*4+1),92 are_congruent_mod 420 &
          92*92*92*92*2|^0 = 420*170569+316 by LemS,Lm1;
        then 2|^(105*4+0),316 are_congruent_mod 420 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 421;
        hence thesis by XPRIMES1:421;
      end;
      suppose A: n = 422;
        2|^6 = 422*0+64 & 64*64*64*64*2|^2 = 422*159025+314 by Lm6,Lm2;
        then 2|^(6*4+2),314 are_congruent_mod 422 &
          314*314*314*314*2|^1 = 422*46071901+210 by Lem1;
        then 2|^(26*4+1),210 are_congruent_mod 422 &
          210*210*210*210*2|^2 = 422*18434218+4 by LemS,Lm2;
        then 2|^(105*4+2),4 are_congruent_mod 422 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 423;
        2|^6 = 423*0+64 & 64*64*64*64*2|^2 = 423*158649+337 by Lm6,Lm2;
        then 2|^(6*4+2),337 are_congruent_mod 423 &
          337*337*337*337*2|^1 = 423*60983062+296 by Lem1;
        then 2|^(26*4+1),296 are_congruent_mod 423 &
          296*296*296*296*2|^3 = 423*145183233+89 by LemS,Lm3;
        then 2|^(105*4+3),89 are_congruent_mod 423 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 424;
        2|^6 = 424*0+64 & 64*64*64*64*2|^2 = 424*158275+264 by Lm6,Lm2;
        then 2|^(6*4+2),264 are_congruent_mod 424 &
          264*264*264*264*2|^2 = 424*45825777+216 by Lem1;
        then 2|^(26*4+2),216 are_congruent_mod 424 &
          216*216*216*216*2|^0 = 424*5133920+256 by LemS,Lm1;
        then 2|^(106*4+0),256 are_congruent_mod 424 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 425;
        2|^6 = 425*0+64 & 64*64*64*64*2|^2 = 425*157903+89 by Lm6,Lm2;
        then 2|^(6*4+2),89 are_congruent_mod 425 &
          89*89*89*89*2|^2 = 425*590515+89 by Lem1;
        then 2|^(26*4+2),89 are_congruent_mod 425 &
          89*89*89*89*2|^1 = 425*295257+257 by LemS;
        then 2|^(106*4+1),257 are_congruent_mod 425 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 426;
        2|^6 = 426*0+64 & 64*64*64*64*2|^2 = 426*157532+232 by Lm6,Lm2;
        then 2|^(6*4+2),232 are_congruent_mod 426 &
          232*232*232*232*2|^2 = 426*27202093+286 by Lem1;
        then 2|^(26*4+2),286 are_congruent_mod 426 &
          286*286*286*286*2|^2 = 426*62822400+64 by LemS;
        then 2|^(106*4+2),64 are_congruent_mod 426 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 427;
        2|^6 = 427*0+64 & 64*64*64*64*2|^2 = 427*157163+263 by Lm6,Lm2;
        then 2|^(6*4+2),263 are_congruent_mod 427 &
          263*263*263*263*2|^2 = 427*44818272+100 by Lem1;
        then 2|^(26*4+2),100 are_congruent_mod 427 &
          100*100*100*100*2|^3 = 427*1873536+128 by LemS,Lm3;
        then 2|^(106*4+3),128 are_congruent_mod 427 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 428;
        2|^6 = 428*0+64 & 64*64*64*64*2|^2 = 428*156796+176 by Lm6,Lm2;
        then 2|^(6*4+2),176 are_congruent_mod 428 &
          176*176*176*176*2|^3 = 428*17934814+216 by Lem1,Lm3;
        then 2|^(26*4+3),216 are_congruent_mod 428 &
          216*216*216*216*2|^0 = 428*5085940+16 by LemS,Lm1;
        then 2|^(107*4+0),16 are_congruent_mod 428 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 429;
        2|^6 = 429*0+64 & 64*64*64*64*2|^2 = 429*156430+394 by Lm6,Lm2;
        then 2|^(6*4+2),394 are_congruent_mod 429 &
          394*394*394*394*2|^3 = 429*449383975+293 by Lem1,Lm3;
        then 2|^(26*4+3),293 are_congruent_mod 429 &
          293*293*293*293*2|^1 = 429*34359211+83 by LemS;
        then 2|^(107*4+1),83 are_congruent_mod 429 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 430;
        2|^6 = 430*0+64 & 64*64*64*64*2|^2 = 430*156067+54 by Lm6,Lm2;
        then 2|^(6*4+2),54 are_congruent_mod 430 &
          54*54*54*54*2|^3 = 430*158196+168 by Lem1,Lm3;
        then 2|^(26*4+3),168 are_congruent_mod 430 &
          168*168*168*168*2|^2 = 430*7410178+164 by LemS,Lm2;
        then 2|^(107*4+2),164 are_congruent_mod 430 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 431;
        hence thesis by XPRIMES1:431;
      end;
      suppose A: n = 432;
        2|^6 = 432*0+64 & 64*64*64*64*2|^3 = 432*310689+80 by Lm6,Lm3;
        then 2|^(6*4+3),80 are_congruent_mod 432 &
          80*80*80*80*2|^0 = 432*94814+352 by Lem1,Lm1;
        then 2|^(27*4+0),352 are_congruent_mod 432 &
          352*352*352*352*2|^0 = 432*35537502+352 by LemS;
        then 2|^(108*4+0),352 are_congruent_mod 432 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 433;
        hence thesis by XPRIMES1:433;
      end;
      suppose A: n = 434;
        2|^6 = 434*0+64 & 64*64*64*64*2|^3 = 434*309257+190 by Lm6,Lm3;
        then 2|^(6*4+3),190 are_congruent_mod 434 &
          190*190*190*190*2|^0 = 434*3002788+8 by Lem1,Lm1;
        then 2|^(27*4+0),8 are_congruent_mod 434 &
          8*8*8*8*2|^2 = 434*37+326 by LemS,Lm2;
        then 2|^(108*4+2),326 are_congruent_mod 434 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 435;
        2|^6 = 435*0+64 & 64*64*64*64*2|^3 = 435*308546+218 by Lm6,Lm3;
        then 2|^(6*4+3),218 are_congruent_mod 435 &
          218*218*218*218*2|^0 = 435*5192024+136 by Lem1,Lm1;
        then 2|^(27*4+0),136 are_congruent_mod 435 &
          136*136*136*136*2|^3 = 435*6291531+143 by LemS,Lm3;
        then 2|^(108*4+3),143 are_congruent_mod 435 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 436;
        2|^6 = 436*0+64 & 64*64*64*64*2|^3 = 436*307838+360 by Lm6,Lm3;
        then 2|^(6*4+3),360 are_congruent_mod 436 &
          360*360*360*360*2|^1 = 436*77046605+220 by Lem1;
        then 2|^(27*4+1),220 are_congruent_mod 436 &
          220*220*220*220*2|^0 = 436*5372844+16 by LemS,Lm1;
        then 2|^(109*4+0),16 are_congruent_mod 436 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 437;
        2|^6 = 437*0+64 & 64*64*64*64*2|^3 = 437*307134+170 by Lm6,Lm3;
        then 2|^(6*4+3),170 are_congruent_mod 437 &
          170*170*170*170*2|^1 = 437*3822471+173 by Lem1;
        then 2|^(27*4+1),173 are_congruent_mod 437 &
          173*173*173*173*2|^1 = 437*4099519+279 by LemS;
        then 2|^(109*4+1),279 are_congruent_mod 437 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 438;
        2|^6 = 438*0+64 & 64*64*64*64*2|^3 = 438*306433+74 by Lm6,Lm3;
        then 2|^(6*4+3),74 are_congruent_mod 438 &
          74*74*74*74*2|^1 = 438*136925+2 by Lem1;
        then 2|^(27*4+1),2 are_congruent_mod 438 &
          2*2*2*2*2|^2 = 438*0+64 by LemS,Lm2;
        then 2|^(109*4+2),64 are_congruent_mod 438 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 439;
        hence thesis by XPRIMES1:439;
      end;
      suppose A: n = 440;
        2|^6 = 440*0+64 & 64*64*64*64*2|^3 = 440*305040+128 by Lm6,Lm3;
        then 2|^(6*4+3),128 are_congruent_mod 440 &
          128*128*128*128*2|^2 = 440*2440322+144 by Lem1,Lm2;
        then 2|^(27*4+2),144 are_congruent_mod 440 &
          144*144*144*144*2|^0 = 440*977231+56 by LemS,Lm1;
        then 2|^(110*4+0),56 are_congruent_mod 440 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 441;
        2|^6 = 441*0+64 & 64*64*64*64*2|^3 = 441*304348+260 by Lm6,Lm3;
        then 2|^(6*4+3),260 are_congruent_mod 441 &
          260*260*260*260*2|^2 = 441*41449070+130 by Lem1,Lm2;
        then 2|^(27*4+2),130 are_congruent_mod 441 &
          130*130*130*130*2|^1 = 441*1295283+197 by LemS;
        then 2|^(110*4+1),197 are_congruent_mod 441 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 442;
        2|^6 = 442*0+64 & 64*64*64*64*2|^3 = 442*303660+8 by Lm6,Lm3;
        then 2|^(6*4+3),8 are_congruent_mod 442 &
          8*8*8*8*2|^2 = 442*37+30 by Lem1,Lm2;
        then 2|^(27*4+2),30 are_congruent_mod 442 &
          30*30*30*30*2|^2 = 442*7330+140 by LemS;
        then 2|^(110*4+2),140 are_congruent_mod 442 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 443;
        hence thesis by XPRIMES1:443;
      end;
      suppose A: n = 444;
        2|^6 = 444*0+64 & 64*64*64*64*2|^3 = 444*302292+80 by Lm6,Lm3;
        then 2|^(6*4+3),80 are_congruent_mod 444 &
          80*80*80*80*2|^3 = 444*738018+8 by Lem1;
        then 2|^(27*4+3),8 are_congruent_mod 444 &
          8*8*8*8*2|^0 = 444*9+100 by LemS,Lm1;
        then 2|^(111*4+0),100 are_congruent_mod 444 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 445;
        2|^6 = 445*0+64 & 64*64*64*64*2|^3 = 445*301612+388 by Lm6,Lm3;
        then 2|^(6*4+3),388 are_congruent_mod 445 &
          388*388*388*388*2|^3 = 445*407433634+358 by Lem1;
        then 2|^(27*4+3),358 are_congruent_mod 445 &
          358*358*358*358*2|^1 = 445*73824768+32 by LemS;
        then 2|^(111*4+1),32 are_congruent_mod 445 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 446;
        2|^6 = 446*0+64 & 64*64*64*64*2|^3 = 446*300936+272 by Lm6,Lm3;
        then 2|^(6*4+3),272 are_congruent_mod 446 &
          272*272*272*272*2|^3 = 446*98181744+224 by Lem1;
        then 2|^(27*4+3),224 are_congruent_mod 446 &
          224*224*224*224*2|^2 = 446*22579650+4 by LemS,Lm2;
        then 2|^(111*4+2),4 are_congruent_mod 446 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 447;
        2|^6 = 447*0+64 & 64*64*64*64*2|^3 = 447*300263+167 by Lm6,Lm3;
        then 2|^(6*4+3),167 are_congruent_mod 447 &
          167*167*167*167*2|^3 = 447*13920292+44 by Lem1;
        then 2|^(27*4+3),44 are_congruent_mod 447 &
          44*44*44*44*2|^3 = 447*67080+8 by LemS;
        then 2|^(111*4+3),8 are_congruent_mod 447 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 448;
        2|^7 = 448*0+128 & 128*128*128*128*2|^0 = 448*599186+128 by Lm7,Lm1;
        then 2|^(7*4+0),128 are_congruent_mod 448 &
          128*128*128*128*2|^0 = 448*599186+128 by Lem1;
        then 2|^(28*4+0),128 are_congruent_mod 448 &
          128*128*128*128*2|^0 = 448*599186+128 by LemS;
        then 2|^(112*4+0),128 are_congruent_mod 448 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 449;
        hence thesis by XPRIMES1:449;
      end;
      suppose A: n = 450;
        2|^7 = 450*0+128 & 128*128*128*128*2|^0 = 450*596523+106 by Lm7,Lm1;
        then 2|^(7*4+0),106 are_congruent_mod 450 &
          106*106*106*106*2|^0 = 450*280550+196 by Lem1;
        then 2|^(28*4+0),196 are_congruent_mod 450 &
          196*196*196*196*2|^2 = 450*13118124+424 by LemS,Lm2;
        then 2|^(112*4+2),424 are_congruent_mod 450 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 451;
        2|^7 = 451*0+128 & 128*128*128*128*2|^0 = 451*595200+256 by Lm7,Lm1;
        then 2|^(7*4+0),256 are_congruent_mod 451 &
          256*256*256*256*2|^0 = 451*9523209+37 by Lem1;
        then 2|^(28*4+0),37 are_congruent_mod 451 &
          37*37*37*37*2|^3 = 451*33244+244 by LemS,Lm3;
        then 2|^(112*4+3),244 are_congruent_mod 451 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 452;
        2|^7 = 452*0+128 & 128*128*128*128*2|^0 = 452*593883+340 by Lm7,Lm1;
        then 2|^(7*4+0),340 are_congruent_mod 452 &
          340*340*340*340*2|^1 = 452*59129911+228 by Lem1;
        then 2|^(28*4+1),228 are_congruent_mod 452 &
          228*228*228*228*2|^0 = 452*5978620+16 by LemS,Lm1;
        then 2|^(113*4+0),16 are_congruent_mod 452 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 453;
        2|^7 = 453*0+128 & 128*128*128*128*2|^0 = 453*592572+340 by Lm7,Lm1;
        then 2|^(7*4+0),340 are_congruent_mod 453 &
          340*340*340*340*2|^1 = 453*58999381+407 by Lem1;
        then 2|^(28*4+1),407 are_congruent_mod 453 &
          407*407*407*407*2|^1 = 453*121146098+8 by LemS;
        then 2|^(113*4+1),8 are_congruent_mod 453 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 454;
        2|^7 = 454*0+128 & 128*128*128*128*2|^0 = 454*591267+238 by Lm7,Lm1;
        then 2|^(7*4+0),238 are_congruent_mod 454 &
          238*238*238*238*2|^1 = 454*14134549+226 by Lem1;
        then 2|^(28*4+1),226 are_congruent_mod 454 &
          226*226*226*226*2|^2 = 454*22984650+4 by LemS,Lm2;
        then 2|^(113*4+2),4 are_congruent_mod 454 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 455;
        2|^7 = 455*0+128 & 128*128*128*128*2|^0 = 455*589968+16 by Lm7,Lm1;
        then 2|^(7*4+0),16 are_congruent_mod 455 &
          16*16*16*16*2|^1 = 455*288+32 by Lem1;
        then 2|^(28*4+1),32 are_congruent_mod 455 &
          32*32*32*32*2|^3 = 455*18436+228 by LemS,Lm3;
        then 2|^(113*4+3),228 are_congruent_mod 455 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 456;
        2|^7 = 456*0+128 & 128*128*128*128*2|^0 = 456*588674+112 by Lm7,Lm1;
        then 2|^(7*4+0),112 are_congruent_mod 456 &
          112*112*112*112*2|^2 = 456*1380280+64 by Lem1,Lm2;
        then 2|^(28*4+2),64 are_congruent_mod 456 &
          64*64*64*64*2|^0 = 456*36792+64 by LemS,Lm1;
        then 2|^(114*4+0),64 are_congruent_mod 456 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 457;
        hence thesis by XPRIMES1:457;
      end;
      suppose A: n = 458;
        2|^7 = 458*0+128 & 128*128*128*128*2|^0 = 458*586103+282 by Lm7,Lm1;
        then 2|^(7*4+0),282 are_congruent_mod 458 &
          282*282*282*282*2|^2 = 458*55232022+228 by Lem1,Lm2;
        then 2|^(28*4+2),228 are_congruent_mod 458 &
          228*228*228*228*2|^2 = 458*23601190+4 by LemS;
        then 2|^(114*4+2),4 are_congruent_mod 458 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 459;
        2|^7 = 459*0+128 & 128*128*128*128*2|^0 = 459*584826+322 by Lm7,Lm1;
        then 2|^(7*4+0),322 are_congruent_mod 459 &
          322*322*322*322*2|^2 = 459*93685157+361 by Lem1,Lm2;
        then 2|^(28*4+2),361 are_congruent_mod 459 &
          361*361*361*361*2|^3 = 459*296009813+161 by LemS,Lm3;
        then 2|^(114*4+3),161 are_congruent_mod 459 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 460;
        2|^7 = 460*0+128 & 128*128*128*128*2|^0 = 460*583555+156 by Lm7,Lm1;
        then 2|^(7*4+0),156 are_congruent_mod 460 &
          156*156*156*156*2|^3 = 460*10299841+308 by Lem1,Lm3;
        then 2|^(28*4+3),308 are_congruent_mod 460 &
          308*308*308*308*2|^0 = 460*19563431+236 by LemS,Lm1;
        then 2|^(115*4+0),236 are_congruent_mod 460 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 461;
        hence thesis by XPRIMES1:461;
      end;
      suppose A: n = 462;
        2|^7 = 462*0+128 & 128*128*128*128*2|^0 = 462*581029+58 by Lm7,Lm1;
        then 2|^(7*4+0),58 are_congruent_mod 462 &
          58*58*58*58*2|^3 = 462*195956+296 by Lem1,Lm3;
        then 2|^(28*4+3),296 are_congruent_mod 462 &
          296*296*296*296*2|^2 = 462*66463752+400 by LemS,Lm2;
        then 2|^(115*4+2),400 are_congruent_mod 462 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 463;
        hence thesis by XPRIMES1:463;
      end;
      suppose A: n = 464;
        2|^7 = 464*0+128 & 128*128*128*128*2|^1 = 464*1157049+176 by Lm7;
        then 2|^(7*4+1),176 are_congruent_mod 464 &
          176*176*176*176*2|^0 = 464*2067915+16 by Lem1,Lm1;
        then 2|^(29*4+0),16 are_congruent_mod 464 &
          16*16*16*16*2|^0 = 464*141+112 by LemS;
        then 2|^(116*4+0),112 are_congruent_mod 464 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 465;
        2|^7 = 465*0+128 & 128*128*128*128*2|^1 = 465*1154561+47 by Lm7;
        then 2|^(7*4+1),47 are_congruent_mod 465 &
          47*47*47*47*2|^0 = 465*10493+436 by Lem1,Lm1;
        then 2|^(29*4+0),436 are_congruent_mod 465 &
          436*436*436*436*2|^1 = 465*155425760+32 by LemS;
        then 2|^(116*4+1),32 are_congruent_mod 465 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 466;
        2|^7 = 466*0+128 & 128*128*128*128*2|^1 = 466*1152083+234 by Lm7;
        then 2|^(7*4+1),234 are_congruent_mod 466 &
          234*234*234*234*2|^0 = 466*6433947+234 by Lem1,Lm1;
        then 2|^(29*4+0),234 are_congruent_mod 466 &
          234*234*234*234*2|^2 = 466*25735790+4 by LemS,Lm2;
        then 2|^(116*4+2),4 are_congruent_mod 466 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 467;
        hence thesis by XPRIMES1:467;
      end;
      suppose A: n = 468;
        2|^7 = 468*0+128 & 128*128*128*128*2|^1 = 468*1147160+32 by Lm7;
        then 2|^(7*4+1),32 are_congruent_mod 468 &
          32*32*32*32*2|^1 = 468*4481+44 by Lem1;
        then 2|^(29*4+1),44 are_congruent_mod 468 &
          44*44*44*44*2|^0 = 468*8008+352 by LemS,Lm1;
        then 2|^(117*4+0),352 are_congruent_mod 468 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 469;
        2|^7 = 469*0+128 & 128*128*128*128*2|^1 = 469*1144714+46 by Lm7;
        then 2|^(7*4+1),46 are_congruent_mod 469 &
          46*46*46*46*2|^1 = 469*19093+295 by Lem1;
        then 2|^(29*4+1),295 are_congruent_mod 469 &
          295*295*295*295*2|^1 = 469*32295738+128 by LemS;
        then 2|^(117*4+1),128 are_congruent_mod 469 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 470;
        2|^7 = 470*0+128 & 128*128*128*128*2|^1 = 470*1142278+252 by Lm7;
        then 2|^(7*4+1),252 are_congruent_mod 470 &
          252*252*252*252*2|^1 = 470*17160672+192 by Lem1;
        then 2|^(29*4+1),192 are_congruent_mod 470 &
          192*192*192*192*2|^2 = 470*11565570+84 by LemS,Lm2;
        then 2|^(117*4+2),84 are_congruent_mod 470 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 471;
        2|^7 = 471*0+128 & 128*128*128*128*2|^1 = 471*1139853+149 by Lm7;
        then 2|^(7*4+1),149 are_congruent_mod 471 &
          149*149*149*149*2|^1 = 471*2092927+185 by Lem1;
        then 2|^(29*4+1),185 are_congruent_mod 471 &
          185*185*185*185*2|^3 = 471*19895552+8 by LemS,Lm3;
        then 2|^(117*4+3),8 are_congruent_mod 471 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 472;
        2|^7 = 472*0+128 & 128*128*128*128*2|^1 = 472*1137438+176 by Lm7;
        then 2|^(7*4+1),176 are_congruent_mod 472 &
          176*176*176*176*2|^2 = 472*8131462+240 by Lem1,Lm2;
        then 2|^(29*4+2),240 are_congruent_mod 472 &
          240*240*240*240*2|^0 = 472*7029152+256 by LemS,Lm1;
        then 2|^(118*4+0),256 are_congruent_mod 472 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 473;
        2|^7 = 473*0+128 & 128*128*128*128*2|^1 = 473*1135033+303 by Lm7;
        then 2|^(7*4+1),303 are_congruent_mod 473 &
          303*303*303*303*2|^2 = 473*71280274+322 by Lem1,Lm2;
        then 2|^(29*4+2),322 are_congruent_mod 473 &
          322*322*322*322*2|^1 = 473*45456117+371 by LemS;
        then 2|^(118*4+1),371 are_congruent_mod 473 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 474;
        2|^7 = 474*0+128 & 128*128*128*128*2|^1 = 474*1132639+26 by Lm7;
        then 2|^(7*4+1),26 are_congruent_mod 474 &
          26*26*26*26*2|^2 = 474*3856+160 by Lem1,Lm2;
        then 2|^(29*4+2),160 are_congruent_mod 474 &
          160*160*160*160*2|^2 = 474*5530464+64 by LemS;
        then 2|^(118*4+2),64 are_congruent_mod 474 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 475;
        2|^7 = 475*0+128 & 128*128*128*128*2|^1 = 475*1130254+262 by Lm7;
        then 2|^(7*4+1),262 are_congruent_mod 475 &
          262*262*262*262*2|^2 = 475*39679989+169 by Lem1,Lm2;
        then 2|^(29*4+2),169 are_congruent_mod 475 &
          169*169*169*169*2|^3 = 475*13738622+318 by LemS,Lm3;
        then 2|^(118*4+3),318 are_congruent_mod 475 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 476;
        2|^7 = 476*0+128 & 128*128*128*128*2|^1 = 476*1127880+32 by Lm7;
        then 2|^(7*4+1),32 are_congruent_mod 476 &
          32*32*32*32*2|^3 = 476*17623+60 by Lem1,Lm3;
        then 2|^(29*4+3),60 are_congruent_mod 476 &
          60*60*60*60*2|^0 = 476*27226+424 by LemS,Lm1;
        then 2|^(119*4+0),424 are_congruent_mod 476 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 477;
        2|^7 = 477*0+128 & 128*128*128*128*2|^1 = 477*1125515+257 by Lm7;
        then 2|^(7*4+1),257 are_congruent_mod 477 &
          257*257*257*257*2|^3 = 477*73165122+14 by Lem1,Lm3;
        then 2|^(29*4+3),14 are_congruent_mod 477 &
          14*14*14*14*2|^1 = 477*161+35 by LemS;
        then 2|^(119*4+1),35 are_congruent_mod 477 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 478;
        2|^7 = 478*0+128 & 128*128*128*128*2|^1 = 478*1123160+432 by Lm7;
        then 2|^(7*4+1),432 are_congruent_mod 478 &
          432*432*432*432*2|^3 = 478*582904056+240 by Lem1,Lm3;
        then 2|^(29*4+3),240 are_congruent_mod 478 &
          240*240*240*240*2|^2 = 478*27763682+4 by LemS,Lm2;
        then 2|^(119*4+2),4 are_congruent_mod 478 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 479;
        hence thesis by XPRIMES1:479;
      end;
      suppose A: n = 480;
        2|^7 = 480*0+128 & 128*128*128*128*2|^2 = 480*2236962+64 by Lm7,Lm2;
        then 2|^(7*4+2),64 are_congruent_mod 480 &
          64*64*64*64*2|^0 = 480*34952+256 by Lem1,Lm1;
        then 2|^(30*4+0),256 are_congruent_mod 480 &
          256*256*256*256*2|^0 = 480*8947848+256 by LemS;
        then 2|^(120*4+0),256 are_congruent_mod 480 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 481;
        2|^7 = 481*0+128 & 128*128*128*128*2|^2 = 481*2232311+233 by Lm7,Lm2;
        then 2|^(7*4+2),233 are_congruent_mod 481 &
          233*233*233*233*2|^0 = 481*6127433+248 by Lem1,Lm1;
        then 2|^(30*4+0),248 are_congruent_mod 481 &
          248*248*248*248*2|^1 = 481*15728657+15 by LemS;
        then 2|^(120*4+1),15 are_congruent_mod 481 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 482;
        2|^7 = 482*0+128 & 128*128*128*128*2|^2 = 482*2227680+64 by Lm7,Lm2;
        then 2|^(7*4+2),64 are_congruent_mod 482 &
          64*64*64*64*2|^0 = 482*34807+242 by Lem1,Lm1;
        then 2|^(30*4+0),242 are_congruent_mod 482 &
          242*242*242*242*2|^2 = 482*28462590+4 by LemS,Lm2;
        then 2|^(120*4+2),4 are_congruent_mod 482 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 483;
        2|^7 = 483*0+128 & 128*128*128*128*2|^2 = 483*2223067+463 by Lm7,Lm2;
        then 2|^(7*4+2),463 are_congruent_mod 483 &
          463*463*463*463*2|^0 = 483*95142998+127 by Lem1,Lm1;
        then 2|^(30*4+0),127 are_congruent_mod 483 &
          127*127*127*127*2|^3 = 483*4308813+449 by LemS,Lm3;
        then 2|^(120*4+3),449 are_congruent_mod 483 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 484;
        2|^7 = 484*0+128 & 128*128*128*128*2|^2 = 484*2218474+408 by Lm7,Lm2;
        then 2|^(7*4+2),408 are_congruent_mod 484 &
          408*408*408*408*2|^1 = 484*114505220+112 by Lem1;
        then 2|^(30*4+1),112 are_congruent_mod 484 &
          112*112*112*112*2|^0 = 484*325107+148 by LemS,Lm1;
        then 2|^(121*4+0),148 are_congruent_mod 484 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 485;
        2|^7 = 485*0+128 & 128*128*128*128*2|^2 = 485*2213900+324 by Lm7,Lm2;
        then 2|^(7*4+2),324 are_congruent_mod 485 &
          324*324*324*324*2|^1 = 485*45443136+192 by Lem1;
        then 2|^(30*4+1),192 are_congruent_mod 485 &
          192*192*192*192*2|^1 = 485*5603936+32 by LemS;
        then 2|^(121*4+1),32 are_congruent_mod 485 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 486;
        2|^7 = 486*0+128 & 128*128*128*128*2|^2 = 486*2209345+154 by Lm7,Lm2;
        then 2|^(7*4+2),154 are_congruent_mod 486 &
          154*154*154*154*2|^1 = 486*2314603+254 by Lem1;
        then 2|^(30*4+1),254 are_congruent_mod 486 &
          254*254*254*254*2|^2 = 486*34257730+244 by LemS,Lm2;
        then 2|^(121*4+2),244 are_congruent_mod 486 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 487;
        hence thesis by XPRIMES1:487;
      end;
      suppose A: n = 488;
        2|^7 = 488*0+128 & 128*128*128*128*2|^2 = 488*2200290+304 by Lm7,Lm2;
        then 2|^(7*4+2),304 are_congruent_mod 488 &
          304*304*304*304*2|^2 = 488*70005877+248 by Lem1;
        then 2|^(30*4+2),248 are_congruent_mod 488 &
          248*248*248*248*2|^0 = 488*7751520+256 by LemS,Lm1;
        then 2|^(122*4+0),256 are_congruent_mod 488 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 489;
        2|^7 = 489*0+128 & 128*128*128*128*2|^2 = 489*2195791+25 by Lm7,Lm2;
        then 2|^(7*4+2),25 are_congruent_mod 489 &
          25*25*25*25*2|^2 = 489*3195+145 by Lem1;
        then 2|^(30*4+2),145 are_congruent_mod 489 &
          145*145*145*145*2|^1 = 489*1807978+8 by LemS;
        then 2|^(122*4+1),8 are_congruent_mod 489 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 490;
        2|^7 = 490*0+128 & 128*128*128*128*2|^2 = 490*2191309+414 by Lm7,Lm2;
        then 2|^(7*4+2),414 are_congruent_mod 490 &
          414*414*414*414*2|^2 = 490*239808888+144 by Lem1;
        then 2|^(30*4+2),144 are_congruent_mod 490 &
          144*144*144*144*2|^2 = 490*3510054+324 by LemS;
        then 2|^(122*4+2),324 are_congruent_mod 490 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 491;
        hence thesis by XPRIMES1:491;
      end;
      suppose A: n = 492;
        2|^7 = 492*0+128 & 128*128*128*128*2|^2 = 492*2182402+40 by Lm7,Lm2;
        then 2|^(7*4+2),40 are_congruent_mod 492 &
          40*40*40*40*2|^3 = 492*41626+8 by Lem1,Lm3;
        then 2|^(30*4+3),8 are_congruent_mod 492 &
          8*8*8*8*2|^0 = 492*8+160 by LemS,Lm1;
        then 2|^(123*4+0),160 are_congruent_mod 492 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 493;
        2|^7 = 493*0+128 & 128*128*128*128*2|^2 = 493*2177975+149 by Lm7,Lm2;
        then 2|^(7*4+2),149 are_congruent_mod 493 &
          149*149*149*149*2|^3 = 493*7998124+76 by Lem1,Lm3;
        then 2|^(30*4+3),76 are_congruent_mod 493 &
          76*76*76*76*2|^1 = 493*135343+253 by LemS;
        then 2|^(123*4+1),253 are_congruent_mod 493 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 494;
        2|^7 = 494*0+128 & 128*128*128*128*2|^2 = 494*2173566+220 by Lm7,Lm2;
        then 2|^(7*4+2),220 are_congruent_mod 494 &
          220*220*220*220*2|^3 = 494*37936194+164 by Lem1,Lm3;
        then 2|^(30*4+3),164 are_congruent_mod 494 &
          164*164*164*164*2|^2 = 494*5857447+446 by LemS,Lm2;
        then 2|^(123*4+2),446 are_congruent_mod 494 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 495;
        2|^7 = 495*0+128 & 128*128*128*128*2|^2 = 495*2169175+199 by Lm7,Lm2;
        then 2|^(7*4+2),199 are_congruent_mod 495 &
          199*199*199*199*2|^3 = 495*25345280+8 by Lem1,Lm3;
        then 2|^(30*4+3),8 are_congruent_mod 495 &
          8*8*8*8*2|^3 = 495*66+98 by LemS;
        then 2|^(123*4+3),98 are_congruent_mod 495 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 496;
        2|^7 = 496*0+128 & 128*128*128*128*2|^3 = 496*4329604+64 by Lm7,Lm3;
        then 2|^(7*4+3),64 are_congruent_mod 496 &
          64*64*64*64*2|^0 = 496*33825+16 by Lem1,Lm1;
        then 2|^(31*4+0),16 are_congruent_mod 496 &
          16*16*16*16*2|^0 = 496*132+64 by LemS;
        then 2|^(124*4+0),64 are_congruent_mod 496 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 497;
        2|^7 = 497*0+128 & 128*128*128*128*2|^3 = 497*4320892+324 by Lm7,Lm3;
        then 2|^(7*4+3),324 are_congruent_mod 497 &
          324*324*324*324*2|^0 = 497*22172958+450 by Lem1,Lm1;
        then 2|^(31*4+0),450 are_congruent_mod 497 &
          450*450*450*450*2|^1 = 497*165015090+270 by LemS;
        then 2|^(124*4+1),270 are_congruent_mod 497 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 498;
        2|^7 = 498*0+128 & 128*128*128*128*2|^3 = 498*4312216+80 by Lm7,Lm3;
        then 2|^(7*4+3),80 are_congruent_mod 498 &
          80*80*80*80*2|^0 = 498*82248+496 by Lem1,Lm1;
        then 2|^(31*4+0),496 are_congruent_mod 498 &
          496*496*496*496*2|^2 = 498*486135520+64 by LemS,Lm2;
        then 2|^(124*4+2),64 are_congruent_mod 498 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 499;
        hence thesis by XPRIMES1:499;
      end;
      suppose A: n = 500;
        2|^7 = 500*0+128 & 128*128*128*128*2|^3 = 500*4294967+148 by Lm7,Lm3;
        then 2|^(7*4+3),148 are_congruent_mod 500 &
          148*148*148*148*2|^1 = 500*1919140+432 by Lem1;
        then 2|^(31*4+1),432 are_congruent_mod 500 &
          432*432*432*432*2|^0 = 500*69657034+376 by LemS,Lm1;
        then 2|^(125*4+0),376 are_congruent_mod 500 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 501;
        2|^7 = 501*0+128 & 128*128*128*128*2|^3 = 501*4286394+254 by Lm7,Lm3;
        then 2|^(7*4+3),254 are_congruent_mod 501 &
          254*254*254*254*2|^1 = 501*16616024+488 by Lem1;
        then 2|^(31*4+1),488 are_congruent_mod 501 &
          488*488*488*488*2|^1 = 501*226397464+8 by LemS;
        then 2|^(125*4+1),8 are_congruent_mod 501 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 502;
        2|^7 = 502*0+128 & 128*128*128*128*2|^3 = 502*4277855+438 by Lm7,Lm3;
        then 2|^(7*4+3),438 are_congruent_mod 502 &
          438*438*438*438*2|^1 = 502*146629961+250 by Lem1;
        then 2|^(31*4+1),250 are_congruent_mod 502 &
          250*250*250*250*2|^2 = 502*31125498+4 by LemS,Lm2;
        then 2|^(125*4+2),4 are_congruent_mod 502 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 503;
        hence thesis by XPRIMES1:503;
      end;
      suppose A: n = 504;
        2|^7 = 504*0+128 & 128*128*128*128*2|^3 = 504*4260880+128 by Lm7,Lm3;
        then 2|^(7*4+3),128 are_congruent_mod 504 &
          128*128*128*128*2|^2 = 504*2130440+64 by Lem1,Lm2;
        then 2|^(31*4+2),64 are_congruent_mod 504 &
          64*64*64*64*2|^0 = 504*33288+64 by LemS,Lm1;
        then 2|^(126*4+0),64 are_congruent_mod 504 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 504 < n & n <= 604;
      then 504+1 <= n & n <= 604 by NAT_1:13;
      then n = 505 or ... or n = 604;
      then per cases;
      suppose A: n = 505;
        2|^7 = 505*0+128 & 128*128*128*128*2|^3 = 505*4252442+438 by Lm7,Lm3;
        then 2|^(7*4+3),438 are_congruent_mod 505 &
          438*438*438*438*2|^2 = 505*291517784+424 by Lem1,Lm2;
        then 2|^(31*4+2),424 are_congruent_mod 505 &
          424*424*424*424*2|^1 = 505*127997664+32 by LemS;
        then 2|^(126*4+1),32 are_congruent_mod 505 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 506;
        2|^7 = 506*0+128 & 128*128*128*128*2|^3 = 506*4244038+420 by Lm7,Lm3;
        then 2|^(7*4+3),420 are_congruent_mod 506 &
          420*420*420*420*2|^2 = 506*245983873+262 by Lem1,Lm2;
        then 2|^(31*4+2),262 are_congruent_mod 506 &
          262*262*262*262*2|^2 = 506*37249001+438 by LemS;
        then 2|^(126*4+2),438 are_congruent_mod 506 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 507;
        2|^7 = 507*0+128 & 128*128*128*128*2|^3 = 507*4235667+479 by Lm7,Lm3;
        then 2|^(7*4+3),479 are_congruent_mod 507 &
          479*479*479*479*2|^2 = 507*415330749+181 by Lem1,Lm2;
        then 2|^(31*4+2),181 are_congruent_mod 507 &
          181*181*181*181*2|^3 = 507*16935433+437 by LemS,Lm3;
        then 2|^(126*4+3),437 are_congruent_mod 507 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 508;
        2|^7 = 508*0+128 & 128*128*128*128*2|^3 = 508*4227330+8 by Lm7,Lm3;
        then 2|^(7*4+3),8 are_congruent_mod 508 &
          8*8*8*8*2|^3 = 508*64+256 by Lem1;
        then 2|^(31*4+3),256 are_congruent_mod 508 &
          256*256*256*256*2|^0 = 508*8454660+16 by LemS,Lm1;
        then 2|^(127*4+0),16 are_congruent_mod 508 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 509;
        hence thesis by XPRIMES1:509;
      end;
      suppose A: n = 510;
        2|^7 = 510*0+128 & 128*128*128*128*2|^3 = 510*4210752+128 by Lm7,Lm3;
        then 2|^(7*4+3),128 are_congruent_mod 510 &
          128*128*128*128*2|^3 = 510*4210752+128 by Lem1;
        then 2|^(31*4+3),128 are_congruent_mod 510 &
          128*128*128*128*2|^2 = 510*2105376+64 by LemS,Lm2;
        then 2|^(127*4+2),64 are_congruent_mod 510 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 511;
        2|^7 = 511*0+128 & 128*128*128*128*2|^3 = 511*4202512+16 by Lm7,Lm3;
        then 2|^(7*4+3),16 are_congruent_mod 511 &
          16*16*16*16*2|^3 = 511*1026+2 by Lem1;
        then 2|^(31*4+3),2 are_congruent_mod 511 &
          2*2*2*2*2|^3 = 511*0+128 by LemS;
        then 2|^(127*4+3),128 are_congruent_mod 511 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 512;
        2|^8 = 512*0+256 & 256*256*256*256*2|^0 = 512*8388608+0 by Lm8,Lm1;
        then 2|^(8*4+0),0 are_congruent_mod 512 &
          0*0*0*0*2|^0 = 512*0+0 by Lem1;
        then 2|^(32*4+0),0 are_congruent_mod 512 &
          0*0*0*0*2|^0 = 512*0+0 by LemS;
        then 2|^(128*4+0),0 are_congruent_mod 512 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 513;
        2|^8 = 513*0+256 & 256*256*256*256*2|^0 = 513*8372255+481 by Lm8,Lm1;
        then 2|^(8*4+0),481 are_congruent_mod 513 &
          481*481*481*481*2|^0 = 513*104342909+4 by Lem1;
        then 2|^(32*4+0),4 are_congruent_mod 513 &
          4*4*4*4*2|^1 = 513*0+512 by LemS;
        then 2|^(128*4+1),512 are_congruent_mod 513 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 514;
        2|^8 = 514*0+256 & 256*256*256*256*2|^0 = 514*8355967+258 by Lm8,Lm1;
        then 2|^(8*4+0),258 are_congruent_mod 514 &
          258*258*258*258*2|^0 = 514*8620167+258 by Lem1;
        then 2|^(32*4+0),258 are_congruent_mod 514 &
          258*258*258*258*2|^2 = 514*34480670+4 by LemS,Lm2;
        then 2|^(128*4+2),4 are_congruent_mod 514 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 515;
        2|^8 = 515*0+256 & 256*256*256*256*2|^0 = 515*8339742+166 by Lm8,Lm1;
        then 2|^(8*4+0),166 are_congruent_mod 515 &
          166*166*166*166*2|^0 = 515*1474433+141 by Lem1;
        then 2|^(32*4+0),141 are_congruent_mod 515 &
          141*141*141*141*2|^3 = 515*6139870+238 by LemS,Lm3;
        then 2|^(128*4+3),238 are_congruent_mod 515 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 516;
        2|^8 = 516*0+256 & 256*256*256*256*2|^0 = 516*8323580+16 by Lm8,Lm1;
        then 2|^(8*4+0),16 are_congruent_mod 516 &
          16*16*16*16*2|^1 = 516*254+8 by Lem1;
        then 2|^(32*4+1),8 are_congruent_mod 516 &
          8*8*8*8*2|^0 = 516*7+484 by LemS,Lm1;
        then 2|^(129*4+0),484 are_congruent_mod 516 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 517;
        2|^8 = 517*0+256 & 256*256*256*256*2|^0 = 517*8307480+136 by Lm8,Lm1;
        then 2|^(8*4+0),136 are_congruent_mod 517 &
          136*136*136*136*2|^1 = 517*1323412+28 by Lem1;
        then 2|^(32*4+1),28 are_congruent_mod 517 &
          28*28*28*28*2|^1 = 517*2377+403 by LemS;
        then 2|^(129*4+1),403 are_congruent_mod 517 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 518;
        2|^8 = 518*0+256 & 256*256*256*256*2|^0 = 518*8291442+340 by Lm8,Lm1;
        then 2|^(8*4+0),340 are_congruent_mod 518 &
          340*340*340*340*2|^1 = 518*51595984+288 by Lem1;
        then 2|^(32*4+1),288 are_congruent_mod 518 &
          288*288*288*288*2|^2 = 518*53125151+326 by LemS,Lm2;
        then 2|^(129*4+2),326 are_congruent_mod 518 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 519;
        2|^8 = 519*0+256 & 256*256*256*256*2|^0 = 519*8275466+442 by Lm8,Lm1;
        then 2|^(8*4+0),442 are_congruent_mod 519 &
          442*442*442*442*2|^1 = 519*147079354+266 by Lem1;
        then 2|^(32*4+1),266 are_congruent_mod 519 &
          266*266*266*266*2|^3 = 519*77170120+8 by LemS,Lm3;
        then 2|^(129*4+3),8 are_congruent_mod 519 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 520;
        2|^8 = 520*0+256 & 256*256*256*256*2|^0 = 520*8259552+256 by Lm8,Lm1;
        then 2|^(8*4+0),256 are_congruent_mod 520 &
          256*256*256*256*2|^2 = 520*33038209+504 by Lem1,Lm2;
        then 2|^(32*4+2),504 are_congruent_mod 520 &
          504*504*504*504*2|^0 = 520*124084862+16 by LemS,Lm1;
        then 2|^(130*4+0),16 are_congruent_mod 520 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 521;
        hence thesis by XPRIMES1:521;
      end;
      suppose A: n = 522;
        2|^8 = 522*0+256 & 256*256*256*256*2|^0 = 522*8227906+364 by Lm8,Lm1;
        then 2|^(8*4+0),364 are_congruent_mod 522 &
          364*364*364*364*2|^2 = 522*134522528+448 by Lem1,Lm2;
        then 2|^(32*4+2),448 are_congruent_mod 522 &
          448*448*448*448*2|^2 = 522*308675062+100 by LemS;
        then 2|^(130*4+2),100 are_congruent_mod 522 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 523;
        hence thesis by XPRIMES1:523;
      end;
      suppose A: n = 524;
        2|^8 = 524*0+256 & 256*256*256*256*2|^0 = 524*8196502+248 by Lm8,Lm1;
        then 2|^(8*4+0),248 are_congruent_mod 524 &
          248*248*248*248*2|^3 = 524*57751786+264 by Lem1,Lm3;
        then 2|^(32*4+3),264 are_congruent_mod 524 &
          264*264*264*264*2|^0 = 524*9270100+16 by LemS,Lm1;
        then 2|^(131*4+0),16 are_congruent_mod 524 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 525;
        2|^8 = 525*0+256 & 256*256*256*256*2|^0 = 525*8180890+46 by Lm8,Lm1;
        then 2|^(8*4+0),46 are_congruent_mod 525 &
          46*46*46*46*2|^3 = 525*68227+473 by Lem1,Lm3;
        then 2|^(32*4+3),473 are_congruent_mod 525 &
          473*473*473*473*2|^1 = 525*190684439+407 by LemS;
        then 2|^(131*4+1),407 are_congruent_mod 525 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 526;
        2|^8 = 526*0+256 & 256*256*256*256*2|^0 = 526*8165337+34 by Lm8,Lm1;
        then 2|^(8*4+0),34 are_congruent_mod 526 &
          34*34*34*34*2|^3 = 526*20324+264 by Lem1,Lm3;
        then 2|^(32*4+3),264 are_congruent_mod 526 &
          264*264*264*264*2|^2 = 526*36939410+4 by LemS,Lm2;
        then 2|^(131*4+2),4 are_congruent_mod 526 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 527;
        2|^8 = 527*0+256 & 256*256*256*256*2|^0 = 527*8149843+35 by Lm8,Lm1;
        then 2|^(8*4+0),35 are_congruent_mod 527 &
          35*35*35*35*2|^3 = 527*22779+467 by Lem1,Lm3;
        then 2|^(32*4+3),467 are_congruent_mod 527 &
          467*467*467*467*2|^3 = 527*722016120+128 by LemS;
        then 2|^(131*4+3),128 are_congruent_mod 527 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 528;
        2|^8 = 528*0+256 & 256*256*256*256*2|^1 = 528*16268815+272 by Lm8;
        then 2|^(8*4+1),272 are_congruent_mod 528 &
          272*272*272*272*2|^0 = 528*10366727+400 by Lem1,Lm1;
        then 2|^(33*4+0),400 are_congruent_mod 528 &
          400*400*400*400*2|^0 = 528*48484848+256 by LemS;
        then 2|^(132*4+0),256 are_congruent_mod 528 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 529;
        2|^8 = 529*0+256 & 256*256*256*256*2|^1 = 529*16238061+323 by Lm8;
        then 2|^(8*4+1),323 are_congruent_mod 529 &
          323*323*323*323*2|^0 = 529*20575690+231 by Lem1,Lm1;
        then 2|^(33*4+0),231 are_congruent_mod 529 &
          231*231*231*231*2|^1 = 529*10765203+255 by LemS;
        then 2|^(132*4+1),255 are_congruent_mod 529 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 530;
        2|^8 = 530*0+256 & 256*256*256*256*2|^1 = 530*16207423+402 by Lm8;
        then 2|^(8*4+1),402 are_congruent_mod 530 &
          402*402*402*402*2|^0 = 530*49275193+526 by Lem1,Lm1;
        then 2|^(33*4+0),526 are_congruent_mod 530 &
          526*526*526*526*2|^2 = 530*577732897+494 by LemS,Lm2;
        then 2|^(132*4+2),494 are_congruent_mod 530 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 531;
        2|^8 = 531*0+256 & 256*256*256*256*2|^1 = 531*16176901+161 by Lm8;
        then 2|^(8*4+1),161 are_congruent_mod 531 &
          161*161*161*161*2|^0 = 531*1265345+46 by Lem1,Lm1;
        then 2|^(33*4+0),46 are_congruent_mod 531 &
          46*46*46*46*2|^3 = 531*67456+512 by LemS,Lm3;
        then 2|^(132*4+3),512 are_congruent_mod 531 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 532;
        2|^8 = 532*0+256 & 256*256*256*256*2|^1 = 532*16146493+316 by Lm8;
        then 2|^(8*4+1),316 are_congruent_mod 532 &
          316*316*316*316*2|^1 = 532*37485792+128 by Lem1;
        then 2|^(33*4+1),128 are_congruent_mod 532 &
          128*128*128*128*2|^0 = 532*504577+492 by LemS,Lm1;
        then 2|^(133*4+0),492 are_congruent_mod 532 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 533;
        2|^8 = 533*0+256 & 256*256*256*256*2|^1 = 533*16116199+525 by Lm8;
        then 2|^(8*4+1),525 are_congruent_mod 533 &
          525*525*525*525*2|^1 = 533*285062441+197 by Lem1;
        then 2|^(33*4+1),197 are_congruent_mod 533 &
          197*197*197*197*2|^1 = 533*5651551+279 by LemS;
        then 2|^(133*4+1),279 are_congruent_mod 533 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 534;
        2|^8 = 534*0+256 & 256*256*256*256*2|^1 = 534*16086019+446 by Lm8;
        then 2|^(8*4+1),446 are_congruent_mod 534 &
          446*446*446*446*2|^1 = 534*148193165+2 by Lem1;
        then 2|^(33*4+1),2 are_congruent_mod 534 &
          2*2*2*2*2|^2 = 534*0+64 by LemS,Lm2;
        then 2|^(133*4+2),64 are_congruent_mod 534 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 535;
        2|^8 = 535*0+256 & 256*256*256*256*2|^1 = 535*16055952+272 by Lm8;
        then 2|^(8*4+1),272 are_congruent_mod 535 &
          272*272*272*272*2|^1 = 535*20462176+352 by Lem1;
        then 2|^(33*4+1),352 are_congruent_mod 535 &
          352*352*352*352*2|^3 = 535*229565625+353 by LemS,Lm3;
        then 2|^(133*4+3),353 are_congruent_mod 535 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 536;
        2|^8 = 536*0+256 & 256*256*256*256*2|^1 = 536*16025997+200 by Lm8;
        then 2|^(8*4+1),200 are_congruent_mod 536 &
          200*200*200*200*2|^2 = 536*11940298+272 by Lem1,Lm2;
        then 2|^(33*4+2),272 are_congruent_mod 536 &
          272*272*272*272*2|^0 = 536*10212000+256 by LemS,Lm1;
        then 2|^(134*4+0),256 are_congruent_mod 536 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 537;
        2|^8 = 537*0+256 & 256*256*256*256*2|^1 = 537*15996153+431 by Lm8;
        then 2|^(8*4+1),431 are_congruent_mod 537 &
          431*431*431*431*2|^2 = 537*257036492+280 by Lem1,Lm2;
        then 2|^(33*4+2),280 are_congruent_mod 537 &
          280*280*280*280*2|^1 = 537*22892216+8 by LemS;
        then 2|^(134*4+1),8 are_congruent_mod 537 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 538;
        2|^8 = 538*0+256 & 256*256*256*256*2|^1 = 538*15966421+94 by Lm8;
        then 2|^(8*4+1),94 are_congruent_mod 538 &
          94*94*94*94*2|^2 = 538*580482+268 by Lem1,Lm2;
        then 2|^(33*4+2),268 are_congruent_mod 538 &
          268*268*268*268*2|^2 = 538*38354550+4 by LemS;
        then 2|^(134*4+2),4 are_congruent_mod 538 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 539;
        2|^8 = 539*0+256 & 256*256*256*256*2|^1 = 539*15936798+470 by Lm8;
        then 2|^(8*4+1),470 are_congruent_mod 539 &
          470*470*470*470*2|^2 = 539*362128460+60 by Lem1,Lm2;
        then 2|^(33*4+2),60 are_congruent_mod 539 &
          60*60*60*60*2|^3 = 539*192356+116 by LemS,Lm3;
        then 2|^(134*4+3),116 are_congruent_mod 539 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 540;
        2|^8 = 540*0+256 & 256*256*256*256*2|^1 = 540*15907286+152 by Lm8;
        then 2|^(8*4+1),152 are_congruent_mod 540 &
          152*152*152*152*2|^3 = 540*7908071+188 by Lem1,Lm3;
        then 2|^(33*4+3),188 are_congruent_mod 540 &
          188*188*188*188*2|^0 = 540*2313330+136 by LemS,Lm1;
        then 2|^(135*4+0),136 are_congruent_mod 540 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 541;
        hence thesis by XPRIMES1:541;
      end;
      suppose A: n = 542;
        2|^8 = 542*0+256 & 256*256*256*256*2|^1 = 542*15848587+438 by Lm8;
        then 2|^(8*4+1),438 are_congruent_mod 542 &
          438*438*438*438*2|^3 = 542*543234248+272 by Lem1,Lm3;
        then 2|^(33*4+3),272 are_congruent_mod 542 &
          272*272*272*272*2|^2 = 542*40395810+4 by LemS,Lm2;
        then 2|^(135*4+2),4 are_congruent_mod 542 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 543;
        2|^8 = 543*0+256 & 256*256*256*256*2|^1 = 543*15819400+392 by Lm8;
        then 2|^(8*4+1),392 are_congruent_mod 543 &
          392*392*392*392*2|^3 = 543*347883976+200 by Lem1,Lm3;
        then 2|^(33*4+3),200 are_congruent_mod 543 &
          200*200*200*200*2|^3 = 543*23572744+8 by LemS;
        then 2|^(135*4+3),8 are_congruent_mod 543 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 544;
        2|^8 = 544*0+256 & 256*256*256*256*2|^2 = 544*31580641+480 by Lm8,Lm2;
        then 2|^(8*4+2),480 are_congruent_mod 544 &
          480*480*480*480*2|^0 = 544*97581176+256 by Lem1,Lm1;
        then 2|^(34*4+0),256 are_congruent_mod 544 &
          256*256*256*256*2|^0 = 544*7895160+256 by LemS;
        then 2|^(136*4+0),256 are_congruent_mod 544 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 545;
        2|^8 = 545*0+256 & 256*256*256*256*2|^2 = 545*31522695+409 by Lm8,Lm2;
        then 2|^(8*4+2),409 are_congruent_mod 545 &
          409*409*409*409*2|^0 = 545*51344831+66 by Lem1,Lm1;
        then 2|^(34*4+0),66 are_congruent_mod 545 &
          66*66*66*66*2|^1 = 545*69632+32 by LemS;
        then 2|^(136*4+1),32 are_congruent_mod 545 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 546;
        2|^8 = 546*0+256 & 256*256*256*256*2|^2 = 546*31464961+478 by Lm8,Lm2;
        then 2|^(8*4+2),478 are_congruent_mod 546 &
          478*478*478*478*2|^0 = 546*95613440+16 by Lem1,Lm1;
        then 2|^(34*4+0),16 are_congruent_mod 546 &
          16*16*16*16*2|^2 = 546*480+64 by LemS,Lm2;
        then 2|^(136*4+2),64 are_congruent_mod 546 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 547;
        hence thesis by XPRIMES1:547;
      end;
      suppose A: n = 548;
        2|^8 = 548*0+256 & 256*256*256*256*2|^2 = 548*31350126+136 by Lm8,Lm2;
        then 2|^(8*4+2),136 are_congruent_mod 548 &
          136*136*136*136*2|^1 = 548*1248547+276 by Lem1;
        then 2|^(34*4+1),276 are_congruent_mod 548 &
          276*276*276*276*2|^0 = 548*10589020+16 by LemS,Lm1;
        then 2|^(137*4+0),16 are_congruent_mod 548 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 549;
        2|^8 = 549*0+256 & 256*256*256*256*2|^2 = 549*31293022+106 by Lm8,Lm2;
        then 2|^(8*4+2),106 are_congruent_mod 549 &
          106*106*106*106*2|^1 = 549*459918+410 by Lem1;
        then 2|^(34*4+1),410 are_congruent_mod 549 &
          410*410*410*410*2|^1 = 549*102942112+512 by LemS;
        then 2|^(137*4+1),512 are_congruent_mod 549 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 550;
        2|^8 = 550*0+256 & 256*256*256*256*2|^2 = 550*31236125+434 by Lm8,Lm2;
        then 2|^(8*4+2),434 are_congruent_mod 550 &
          434*434*434*434*2|^1 = 550*129010846+172 by Lem1;
        then 2|^(34*4+1),172 are_congruent_mod 550 &
          172*172*172*172*2|^2 = 550*6365185+474 by LemS,Lm2;
        then 2|^(137*4+2),474 are_congruent_mod 550 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 551;
        2|^8 = 551*0+256 & 256*256*256*256*2|^2 = 551*31179435+499 by Lm8,Lm2;
        then 2|^(8*4+2),499 are_congruent_mod 551 &
          499*499*499*499*2|^1 = 551*225050809+243 by Lem1;
        then 2|^(34*4+1),243 are_congruent_mod 551 &
          243*243*243*243*2|^3 = 551*50624818+490 by LemS,Lm3;
        then 2|^(137*4+3),490 are_congruent_mod 551 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 552;
        2|^8 = 552*0+256 & 256*256*256*256*2|^2 = 552*31122951+232 by Lm8,Lm2;
        then 2|^(8*4+2),232 are_congruent_mod 552 &
          232*232*232*232*2|^2 = 552*20992920+64 by Lem1;
        then 2|^(34*4+2),64 are_congruent_mod 552 &
          64*64*64*64*2|^0 = 552*30393+280 by LemS,Lm1;
        then 2|^(138*4+0),280 are_congruent_mod 552 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 553;
        2|^8 = 553*0+256 & 256*256*256*256*2|^2 = 553*31066671+121 by Lm8,Lm2;
        then 2|^(8*4+2),121 are_congruent_mod 553 &
          121*121*121*121*2|^2 = 553*1550516+176 by Lem1;
        then 2|^(34*4+2),176 are_congruent_mod 553 &
          176*176*176*176*2|^1 = 553*3470208+128 by LemS;
        then 2|^(138*4+1),128 are_congruent_mod 553 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 554;
        2|^8 = 554*0+256 & 256*256*256*256*2|^2 = 554*31010594+108 by Lm8,Lm2;
        then 2|^(8*4+2),108 are_congruent_mod 554 &
          108*108*108*108*2|^2 = 554*982302+276 by Lem1;
        then 2|^(34*4+2),276 are_congruent_mod 554 &
          276*276*276*276*2|^2 = 554*41897350+4 by LemS;
        then 2|^(138*4+2),4 are_congruent_mod 554 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 555;
        2|^8 = 555*0+256 & 256*256*256*256*2|^2 = 555*30954719+139 by Lm8,Lm2;
        then 2|^(8*4+2),139 are_congruent_mod 555 &
          139*139*139*139*2|^2 = 555*2690457+529 by Lem1;
        then 2|^(34*4+2),529 are_congruent_mod 555 &
          529*529*529*529*2|^3 = 555*1128806995+23 by LemS,Lm3;
        then 2|^(138*4+3),23 are_congruent_mod 555 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 556;
        2|^8 = 556*0+256 & 256*256*256*256*2|^2 = 556*30899045+164 by Lm8,Lm2;
        then 2|^(8*4+2),164 are_congruent_mod 556 &
          164*164*164*164*2|^3 = 556*10408558+280 by Lem1,Lm3;
        then 2|^(34*4+3),280 are_congruent_mod 556 &
          280*280*280*280*2|^0 = 556*11054964+16 by LemS,Lm1;
        then 2|^(139*4+0),16 are_congruent_mod 556 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 557;
        hence thesis by XPRIMES1:557;
      end;
      suppose A: n = 558;
        2|^8 = 558*0+256 & 256*256*256*256*2|^2 = 558*30788296+16 by Lm8,Lm2;
        then 2|^(8*4+2),16 are_congruent_mod 558 &
          16*16*16*16*2|^3 = 558*939+326 by Lem1,Lm3;
        then 2|^(34*4+3),326 are_congruent_mod 558 &
          326*326*326*326*2|^2 = 558*80964789+442 by LemS,Lm2;
        then 2|^(139*4+2),442 are_congruent_mod 558 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 559;
        2|^8 = 559*0+256 & 256*256*256*256*2|^2 = 559*30733218+322 by Lm8,Lm2;
        then 2|^(8*4+2),322 are_congruent_mod 559 &
          322*322*322*322*2|^3 = 559*153851475+323 by Lem1,Lm3;
        then 2|^(34*4+3),323 are_congruent_mod 559 &
          323*323*323*323*2|^3 = 559*155771595+323 by LemS;
        then 2|^(139*4+3),323 are_congruent_mod 559 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 560;
        2|^8 = 560*0+256 & 256*256*256*256*2|^3 = 560*61356675+368 by Lm8,Lm3;
        then 2|^(8*4+3),368 are_congruent_mod 560 &
          368*368*368*368*2|^0 = 560*32749392+256 by Lem1,Lm1;
        then 2|^(35*4+0),256 are_congruent_mod 560 &
          256*256*256*256*2|^0 = 560*7669584+256 by LemS;
        then 2|^(140*4+0),256 are_congruent_mod 560 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 561;
        hence thesis by A5,ENUMSET1:def 2;
      end;
      suppose A: n = 562;
        2|^8 = 562*0+256 & 256*256*256*256*2|^3 = 562*61138324+280 by Lm8,Lm3;
        then 2|^(8*4+3),280 are_congruent_mod 562 &
          280*280*280*280*2|^0 = 562*10936939+282 by Lem1,Lm1;
        then 2|^(35*4+0),282 are_congruent_mod 562 &
          282*282*282*282*2|^2 = 562*45011150+4 by LemS,Lm2;
        then 2|^(140*4+2),4 are_congruent_mod 562 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 563;
        hence thesis by XPRIMES1:563;
      end;
      suppose A: n = 564;
        2|^8 = 564*0+256 & 256*256*256*256*2|^3 = 564*60921521+524 by Lm8,Lm3;
        then 2|^(8*4+3),524 are_congruent_mod 564 &
          524*524*524*524*2|^1 = 564*267347446+8 by Lem1;
        then 2|^(35*4+1),8 are_congruent_mod 564 &
          8*8*8*8*2|^0 = 564*7+148 by LemS,Lm1;
        then 2|^(141*4+0),148 are_congruent_mod 564 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 565;
        2|^8 = 565*0+256 & 256*256*256*256*2|^3 = 565*60813696+128 by Lm8,Lm3;
        then 2|^(8*4+3),128 are_congruent_mod 565 &
          128*128*128*128*2|^1 = 565*950214+2 by Lem1;
        then 2|^(35*4+1),2 are_congruent_mod 565 &
          2*2*2*2*2|^1 = 565*0+32 by LemS;
        then 2|^(141*4+1),32 are_congruent_mod 565 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 566;
        2|^8 = 566*0+256 & 256*256*256*256*2|^3 = 566*60706251+302 by Lm8,Lm3;
        then 2|^(8*4+3),302 are_congruent_mod 566 &
          302*302*302*302*2|^1 = 566*29392825+282 by Lem1;
        then 2|^(35*4+1),282 are_congruent_mod 566 &
          282*282*282*282*2|^2 = 566*44693050+4 by LemS,Lm2;
        then 2|^(141*4+2),4 are_congruent_mod 566 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 567;
        2|^8 = 567*0+256 & 256*256*256*256*2|^3 = 567*60599185+473 by Lm8,Lm3;
        then 2|^(8*4+3),473 are_congruent_mod 567 &
          473*473*473*473*2|^1 = 567*176559666+260 by Lem1;
        then 2|^(35*4+1),260 are_congruent_mod 567 &
          260*260*260*260*2|^3 = 567*64476331+323 by LemS,Lm3;
        then 2|^(141*4+3),323 are_congruent_mod 567 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 568;
        2|^8 = 568*0+256 & 256*256*256*256*2|^3 = 568*60492497+72 by Lm8,Lm3;
        then 2|^(8*4+3),72 are_congruent_mod 568 &
          72*72*72*72*2|^2 = 568*189252+288 by Lem1,Lm2;
        then 2|^(35*4+2),288 are_congruent_mod 568 &
          288*288*288*288*2|^0 = 568*12112160+256 by LemS,Lm1;
        then 2|^(142*4+0),256 are_congruent_mod 568 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 569;
        hence thesis by XPRIMES1:569;
      end;
      suppose A: n = 570;
        2|^8 = 570*0+256 & 256*256*256*256*2|^3 = 570*60280242+428 by Lm8,Lm3;
        then 2|^(8*4+3),428 are_congruent_mod 570 &
          428*428*428*428*2|^2 = 570*235483353+214 by Lem1,Lm2;
        then 2|^(35*4+2),214 are_congruent_mod 570 &
          214*214*214*214*2|^2 = 570*14717709+334 by LemS;
        then 2|^(142*4+2),334 are_congruent_mod 570 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 571;
        hence thesis by XPRIMES1:571;
      end;
      suppose A: n = 572;
        2|^8 = 572*0+256 & 256*256*256*256*2|^3 = 572*60069472+384 by Lm8,Lm3;
        then 2|^(8*4+3),384 are_congruent_mod 572 &
          384*384*384*384*2|^3 = 572*304101705+228 by Lem1;
        then 2|^(35*4+3),228 are_congruent_mod 572 &
          228*228*228*228*2|^0 = 572*4724364+48 by LemS,Lm1;
        then 2|^(143*4+0),48 are_congruent_mod 572 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 573;
        2|^8 = 573*0+256 & 256*256*256*256*2|^3 = 573*59964639+221 by Lm8,Lm3;
        then 2|^(8*4+3),221 are_congruent_mod 573 &
          221*221*221*221*2|^3 = 573*33304618+134 by Lem1;
        then 2|^(35*4+3),134 are_congruent_mod 573 &
          134*134*134*134*2|^1 = 573*1125368+8 by LemS;
        then 2|^(143*4+1),8 are_congruent_mod 573 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 574;
        2|^8 = 574*0+256 & 256*256*256*256*2|^3 = 574*59860171+214 by Lm8,Lm3;
        then 2|^(8*4+3),214 are_congruent_mod 574 &
          214*214*214*214*2|^3 = 574*29230294+172 by Lem1;
        then 2|^(35*4+3),172 are_congruent_mod 574 &
          172*172*172*172*2|^2 = 574*6099045+394 by LemS,Lm2;
        then 2|^(143*4+2),394 are_congruent_mod 574 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 575;
        2|^8 = 575*0+256 & 256*256*256*256*2|^3 = 575*59756066+418 by Lm8,Lm3;
        then 2|^(8*4+3),418 are_congruent_mod 575 &
          418*418*418*418*2|^3 = 575*424744016+208 by Lem1;
        then 2|^(35*4+3),208 are_congruent_mod 575 &
          208*208*208*208*2|^3 = 575*26042068+468 by LemS;
        then 2|^(143*4+3),468 are_congruent_mod 575 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 576;
        2|^9 = 576*0+512 & 512*512*512*512*2|^0 = 576*119304647+64 by Lm9,Lm1;
        then 2|^(9*4+0),64 are_congruent_mod 576 &
          64*64*64*64*2|^0 = 576*29127+64 by Lem1;
        then 2|^(36*4+0),64 are_congruent_mod 576 &
          64*64*64*64*2|^0 = 576*29127+64 by LemS;
        then 2|^(144*4+0),64 are_congruent_mod 576 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 577;
        hence thesis by XPRIMES1:577;
      end;
      suppose A: n = 578;
        2|^9 = 578*0+512 &
          512*512*512*512*2|^0 = 578*118891828+152 by Lm9,Lm1;
        then 2|^(9*4+0),152 are_congruent_mod 578 &
          152*152*152*152*2|^0 = 578*923520+256 by Lem1;
        then 2|^(36*4+0),256 are_congruent_mod 578 &
          256*256*256*256*2|^2 = 578*29722957+38 by LemS,Lm2;
        then 2|^(144*4+2),38 are_congruent_mod 578 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 579;
        2|^9 = 579*0+512 &
          512*512*512*512*2|^0 = 579*118686488+184 by Lm9,Lm1;
        then 2|^(9*4+0),184 are_congruent_mod 579 &
          184*184*184*184*2|^0 = 579*1979669+385 by Lem1;
        then 2|^(36*4+0),385 are_congruent_mod 579 &
          385*385*385*385*2|^3 = 579*303566848+8 by LemS,Lm3;
        then 2|^(144*4+3),8 are_congruent_mod 579 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 580;
        2|^9 = 580*0+512 &
          512*512*512*512*2|^0 = 580*118481856+256 by Lm9,Lm1;
        then 2|^(9*4+0),256 are_congruent_mod 580 &
          256*256*256*256*2|^1 = 580*14810232+32 by Lem1;
        then 2|^(36*4+1),32 are_congruent_mod 580 &
          32*32*32*32*2|^0 = 580*1807+516 by LemS,Lm1;
        then 2|^(145*4+0),516 are_congruent_mod 580 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 581;
        2|^9 = 581*0+512 &
          512*512*512*512*2|^0 = 581*118277928+568 by Lm9,Lm1;
        then 2|^(9*4+0),568 are_congruent_mod 581 &
          568*568*568*568*2|^1 = 581*358300328+184 by Lem1;
        then 2|^(36*4+1),184 are_congruent_mod 581 &
          184*184*184*184*2|^1 = 581*3945709+543 by LemS;
        then 2|^(145*4+1),543 are_congruent_mod 581 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 582;
        2|^9 = 582*0+512 &
          512*512*512*512*2|^0 = 582*118074702+172 by Lm9,Lm1;
        then 2|^(9*4+0),172 are_congruent_mod 582 &
          172*172*172*172*2|^1 = 582*3007605+2 by Lem1;
        then 2|^(36*4+1),2 are_congruent_mod 582 &
          2*2*2*2*2|^2 = 582*0+64 by LemS,Lm2;
        then 2|^(145*4+2),64 are_congruent_mod 582 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 583;
        2|^9 = 583*0+512 &
          512*512*512*512*2|^0 = 583*117872172+460 by Lm9,Lm1;
        then 2|^(9*4+0),460 are_congruent_mod 583 &
          460*460*460*460*2|^1 = 583*153600548+516 by Lem1;
        then 2|^(36*4+1),516 are_congruent_mod 583 &
          516*516*516*516*2|^3 = 583*972792556+140 by LemS,Lm3;
        then 2|^(145*4+3),140 are_congruent_mod 583 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 584;
        2|^9 = 584*0+512 &
          512*512*512*512*2|^0 = 584*117670336+512 by Lm9,Lm1;
        then 2|^(9*4+0),512 are_congruent_mod 584 &
          512*512*512*512*2|^2 = 584*470681347+296 by Lem1,Lm2;
        then 2|^(36*4+2),296 are_congruent_mod 584 &
          296*296*296*296*2|^0 = 584*13144800+256 by LemS,Lm1;
        then 2|^(146*4+0),256 are_congruent_mod 584 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 585;
        2|^9 = 585*0+512 & 512*512*512*512*2|^0 = 585*117469191+1 by Lm9,Lm1;
        then 2|^(9*4+0),1 are_congruent_mod 585 &
          1*1*1*1*2|^2 = 585*0+4 by Lem1,Lm2;
        then 2|^(36*4+2),4 are_congruent_mod 585 &
          4*4*4*4*2|^1 = 585*0+512 by LemS;
        then 2|^(146*4+1),512 are_congruent_mod 585 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 586;
        2|^9 = 586*0+512 &
          512*512*512*512*2|^0 = 586*117268731+370 by Lm9,Lm1;
        then 2|^(9*4+0),370 are_congruent_mod 586 &
          370*370*370*370*2|^2 = 586*127929078+292 by Lem1,Lm2;
        then 2|^(36*4+2),292 are_congruent_mod 586 &
          292*292*292*292*2|^2 = 586*49624230+4 by LemS;
        then 2|^(146*4+2),4 are_congruent_mod 586 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 587;
        hence thesis by XPRIMES1:587;
      end;
      suppose A: n = 588;
        2|^9 = 588*0+512 &
          512*512*512*512*2|^0 = 588*116869858+232 by Lm9,Lm1;
        then 2|^(9*4+0),232 are_congruent_mod 588 &
          232*232*232*232*2|^3 = 588*39415278+344 by Lem1,Lm3;
        then 2|^(36*4+3),344 are_congruent_mod 588 &
          344*344*344*344*2|^0 = 588*23815321+148 by LemS,Lm1;
        then 2|^(147*4+0),148 are_congruent_mod 588 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 589;
        2|^9 = 589*0+512 &
          512*512*512*512*2|^0 = 589*116671437+343 by Lm9,Lm1;
        then 2|^(9*4+0),343 are_congruent_mod 589 &
          343*343*343*343*2|^3 = 589*187997109+407 by Lem1,Lm3;
        then 2|^(36*4+3),407 are_congruent_mod 589 &
          407*407*407*407*2|^1 = 589*93173484+326 by LemS;
        then 2|^(147*4+1),326 are_congruent_mod 589 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 590;
        2|^9 = 590*0+512 &
          512*512*512*512*2|^0 = 590*116473689+226 by Lm9,Lm1;
        then 2|^(9*4+0),226 are_congruent_mod 590 &
          226*226*226*226*2|^3 = 590*35372986+468 by Lem1,Lm3;
        then 2|^(36*4+3),468 are_congruent_mod 590 &
          468*468*468*468*2|^2 = 590*325230593+434 by LemS,Lm2;
        then 2|^(147*4+2),434 are_congruent_mod 590 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 591;
        2|^9 = 591*0+512 &
          512*512*512*512*2|^0 = 591*116276610+226 by Lm9,Lm1;
        then 2|^(9*4+0),226 are_congruent_mod 591 &
          226*226*226*226*2|^3 = 591*35313134+14 by Lem1,Lm3;
        then 2|^(36*4+3),14 are_congruent_mod 591 &
          14*14*14*14*2|^3 = 591*520+8 by LemS;
        then 2|^(147*4+3),8 are_congruent_mod 591 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 592;
        2|^9 = 592*0+512 &
          512*512*512*512*2|^1 = 592*232160394+224 by Lm9;
        then 2|^(9*4+1),224 are_congruent_mod 592 &
          224*224*224*224*2|^0 = 592*4252755+16 by Lem1,Lm1;
        then 2|^(37*4+0),16 are_congruent_mod 592 &
          16*16*16*16*2|^0 = 592*110+416 by LemS;
        then 2|^(148*4+0),416 are_congruent_mod 592 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 593;
        hence thesis by XPRIMES1:593;
      end;
      suppose A: n = 594;
        2|^9 = 594*0+512 &
          512*512*512*512*2|^1 = 594*231378709+326 by Lm9;
        then 2|^(9*4+1),326 are_congruent_mod 594 &
          326*326*326*326*2|^0 = 594*19014458+124 by Lem1,Lm1;
        then 2|^(37*4+0),124 are_congruent_mod 594 &
          124*124*124*124*2|^2 = 594*1592063+82 by LemS,Lm2;
        then 2|^(148*4+2),82 are_congruent_mod 594 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 595;
        2|^9 = 595*0+512 &
          512*512*512*512*2|^1 = 595*230989837+457 by Lm9;
        then 2|^(9*4+1),457 are_congruent_mod 595 &
          457*457*457*457*2|^0 = 595*73307403+16 by Lem1,Lm1;
        then 2|^(37*4+0),16 are_congruent_mod 595 &
          16*16*16*16*2|^3 = 595*881+93 by LemS,Lm3;
        then 2|^(148*4+3),93 are_congruent_mod 595 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 596;
        2|^9 = 596*0+512 &
          512*512*512*512*2|^1 = 596*230602270+552 by Lm9;
        then 2|^(9*4+1),552 are_congruent_mod 596 &
          552*552*552*552*2|^1 = 596*311558817+300 by Lem1;
        then 2|^(37*4+1),300 are_congruent_mod 596 &
          300*300*300*300*2|^0 = 596*13590604+16 by LemS,Lm1;
        then 2|^(149*4+0),16 are_congruent_mod 596 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 597;
        2|^9 = 597*0+512 &
          512*512*512*512*2|^1 = 597*230216002+278 by Lm9;
        then 2|^(9*4+1),278 are_congruent_mod 597 &
          278*278*278*278*2|^1 = 597*20009436+20 by Lem1;
        then 2|^(37*4+1),20 are_congruent_mod 597 &
          20*20*20*20*2|^1 = 597*536+8 by LemS;
        then 2|^(149*4+1),8 are_congruent_mod 597 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 598;
        2|^9 = 598*0+512 &
          512*512*512*512*2|^1 = 598*229831025+522 by Lm9;
        then 2|^(9*4+1),522 are_congruent_mod 598 &
          522*522*522*522*2|^1 = 598*248319499+110 by Lem1;
        then 2|^(37*4+1),110 are_congruent_mod 598 &
          110*110*110*110*2|^2 = 598*979331+62 by LemS,Lm2;
        then 2|^(149*4+2),62 are_congruent_mod 598 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 599;
        hence thesis by XPRIMES1:599;
      end;
      suppose A: n = 600;
        2|^9 = 600*0+512 &
          512*512*512*512*2|^1 = 600*229064922+272 by Lm9;
        then 2|^(9*4+1),272 are_congruent_mod 600 &
          272*272*272*272*2|^2 = 600*36490881+424 by Lem1,Lm2;
        then 2|^(37*4+2),424 are_congruent_mod 600 &
          424*424*424*424*2|^0 = 600*53865683+376 by LemS,Lm1;
        then 2|^(150*4+0),376 are_congruent_mod 600 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 601;
        hence thesis by XPRIMES1:601;
      end;
      suppose A: n = 602;
        2|^9 = 602*0+512 &
          512*512*512*512*2|^1 = 602*228303909+254 by Lm9;
        then 2|^(9*4+1),254 are_congruent_mod 602 &
          254*254*254*254*2|^2 = 602*27656573+78 by Lem1,Lm2;
        then 2|^(37*4+2),78 are_congruent_mod 602 &
          78*78*78*78*2|^2 = 602*245947+130 by LemS;
        then 2|^(150*4+2),130 are_congruent_mod 602 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 603;
        2|^9 = 603*0+512 &
          512*512*512*512*2|^1 = 603*227925295+587 by Lm9;
        then 2|^(9*4+1),587 are_congruent_mod 603 &
          587*587*587*587*2|^2 = 603*787580734+442 by Lem1,Lm2;
        then 2|^(37*4+2),442 are_congruent_mod 603 &
          442*442*442*442*2|^3 = 603*506362752+512 by LemS,Lm3;
        then 2|^(150*4+3),512 are_congruent_mod 603 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 604;
        2|^9 = 604*0+512 &
          512*512*512*512*2|^1 = 604*227547936+128 by Lm9;
        then 2|^(9*4+1),128 are_congruent_mod 604 &
          128*128*128*128*2|^3 = 604*3555436+304 by Lem1,Lm3;
        then 2|^(37*4+3),304 are_congruent_mod 604 &
          304*304*304*304*2|^0 = 604*14140260+16 by LemS,Lm1;
        then 2|^(151*4+0),16 are_congruent_mod 604 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 604 < n & n <= 704;
      then 604+1 <= n & n <= 704 by NAT_1:13;
      then n = 605 or ... or n = 704;
      then per cases;
      suppose A: n = 605;
        2|^9 = 605*0+512 &
          512*512*512*512*2|^1 = 605*227171823+557 by Lm9;
        then 2|^(9*4+1),557 are_congruent_mod 605 &
          557*557*557*557*2|^3 = 605*1272786009+563 by Lem1,Lm3;
        then 2|^(37*4+3),563 are_congruent_mod 605 &
          563*563*563*563*2|^1 = 605*332130072+362 by LemS;
        then 2|^(151*4+1),362 are_congruent_mod 605 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 606;
        2|^9 = 606*0+512 &
          512*512*512*512*2|^1 = 606*226796952+560 by Lm9;
        then 2|^(9*4+1),560 are_congruent_mod 606 &
          560*560*560*560*2|^3 = 606*1298283300+200 by Lem1,Lm3;
        then 2|^(37*4+3),200 are_congruent_mod 606 &
          200*200*200*200*2|^2 = 606*10561056+64 by LemS,Lm2;
        then 2|^(151*4+2),64 are_congruent_mod 606 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 607;
        hence thesis by XPRIMES1:607;
      end;
      suppose A: n = 608;
        2|^9 = 608*0+512 &
          512*512*512*512*2|^2 = 608*452101820+384 by Lm9,Lm2;
        then 2|^(9*4+2),384 are_congruent_mod 608 &
          384*384*384*384*2|^0 = 608*35761960+256 by Lem1,Lm1;
        then 2|^(38*4+0),256 are_congruent_mod 608 &
          256*256*256*256*2|^0 = 608*7064090+576 by LemS;
        then 2|^(152*4+0),576 are_congruent_mod 608 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 609;
        2|^9 = 609*0+512 & 512*512*512*512*2|^2 = 609*451359453+67 by Lm9,Lm2;
        then 2|^(9*4+2),67 are_congruent_mod 609 &
          67*67*67*67*2|^0 = 609*33088+529 by Lem1,Lm1;
        then 2|^(38*4+0),529 are_congruent_mod 609 &
          529*529*529*529*2|^1 = 609*257178933+365 by LemS;
        then 2|^(152*4+1),365 are_congruent_mod 609 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 610;
        2|^9 = 610*0+512 &
          512*512*512*512*2|^2 = 610*450619519+354 by Lm9,Lm2;
        then 2|^(9*4+2),354 are_congruent_mod 610 &
          354*354*354*354*2|^0 = 610*25744425+606 by Lem1,Lm1;
        then 2|^(38*4+0),606 are_congruent_mod 610 &
          606*606*606*606*2|^2 = 610*884342817+414 by LemS,Lm2;
        then 2|^(152*4+2),414 are_congruent_mod 610 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 611;
        2|^9 = 611*0+512 & 512*512*512*512*2|^2 = 611*449882008+56 by Lm9,Lm2;
        then 2|^(9*4+2),56 are_congruent_mod 611 &
          56*56*56*56*2|^0 = 611*16095+451 by Lem1,Lm1;
        then 2|^(38*4+0),451 are_congruent_mod 611 &
          451*451*451*451*2|^3 = 611*541695146+202 by LemS,Lm3;
        then 2|^(152*4+3),202 are_congruent_mod 611 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 612;
        2|^9 = 612*0+512 &
          512*512*512*512*2|^2 = 612*449146906+472 by Lm9,Lm2;
        then 2|^(9*4+2),472 are_congruent_mod 612 &
          472*472*472*472*2|^1 = 612*162198400+512 by Lem1;
        then 2|^(38*4+1),512 are_congruent_mod 612 &
          512*512*512*512*2|^0 = 612*112286726+424 by LemS,Lm1;
        then 2|^(153*4+0),424 are_congruent_mod 612 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 613;
        hence thesis by XPRIMES1:613;
      end;
      suppose A: n = 614;
        2|^9 = 614*0+512 &
          512*512*512*512*2|^2 = 614*447683887+326 by Lm9,Lm2;
        then 2|^(9*4+2),326 are_congruent_mod 614 &
          326*326*326*326*2|^1 = 614*36790189+306 by Lem1;
        then 2|^(38*4+1),306 are_congruent_mod 614 &
          306*306*306*306*2|^2 = 614*57118570+4 by LemS,Lm2;
        then 2|^(153*4+2),4 are_congruent_mod 614 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 615;
        2|^9 = 615*0+512 &
          512*512*512*512*2|^2 = 615*446955946+154 by Lm9,Lm2;
        then 2|^(9*4+2),154 are_congruent_mod 615 &
          154*154*154*154*2|^1 = 615*1829101+197 by Lem1;
        then 2|^(38*4+1),197 are_congruent_mod 615 &
          197*197*197*197*2|^3 = 615*19592045+173 by LemS,Lm3;
        then 2|^(153*4+3),173 are_congruent_mod 615 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 616;
        2|^9 = 616*0+512 &
          512*512*512*512*2|^2 = 616*446230368+256 by Lm9,Lm2;
        then 2|^(9*4+2),256 are_congruent_mod 616 &
          256*256*256*256*2|^2 = 616*27889398+16 by Lem1;
        then 2|^(38*4+2),16 are_congruent_mod 616 &
          16*16*16*16*2|^0 = 616*106+240 by LemS,Lm1;
        then 2|^(154*4+0),240 are_congruent_mod 616 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 617;
        hence thesis by XPRIMES1:617;
      end;
      suppose A: n = 618;
        2|^9 = 618*0+512 &
          512*512*512*512*2|^2 = 618*444786257+118 by Lm9,Lm2;
        then 2|^(9*4+2),118 are_congruent_mod 618 &
          118*118*118*118*2|^2 = 618*1254872+208 by Lem1;
        then 2|^(38*4+2),208 are_congruent_mod 618 &
          208*208*208*208*2|^2 = 618*12115040+64 by LemS;
        then 2|^(154*4+2),64 are_congruent_mod 618 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 619;
        hence thesis by XPRIMES1:619;
      end;
      suppose A: n = 620;
        2|^9 = 620*0+512 &
          512*512*512*512*2|^2 = 620*443351462+504 by Lm9,Lm2;
        then 2|^(9*4+2),504 are_congruent_mod 620 &
          504*504*504*504*2|^3 = 620*832569396+528 by Lem1,Lm3;
        then 2|^(38*4+3),528 are_congruent_mod 620 &
          528*528*528*528*2|^0 = 620*125355675+156 by LemS,Lm1;
        then 2|^(155*4+0),156 are_congruent_mod 620 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 621;
        2|^9 = 621*0+512 &
          512*512*512*512*2|^2 = 621*442637531+193 by Lm9,Lm2;
        then 2|^(9*4+2),193 are_congruent_mod 621 &
          193*193*193*193*2|^3 = 621*17874241+347 by Lem1,Lm3;
        then 2|^(38*4+3),347 are_congruent_mod 621 &
          347*347*347*347*2|^1 = 621*46693485+377 by LemS;
        then 2|^(155*4+1),377 are_congruent_mod 621 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 622;
        2|^9 = 622*0+512 &
          512*512*512*512*2|^2 = 622*441925895+254 by Lm9,Lm2;
        then 2|^(9*4+2),254 are_congruent_mod 622 &
          254*254*254*254*2|^3 = 622*53534588+312 by Lem1,Lm3;
        then 2|^(38*4+3),312 are_congruent_mod 622 &
          312*312*312*312*2|^2 = 622*60937970+4 by LemS,Lm2;
        then 2|^(155*4+2),4 are_congruent_mod 622 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 623;
        2|^9 = 623*0+512 & 512*512*512*512*2|^2 = 623*441216544+32 by Lm9,Lm2;
        then 2|^(9*4+2),32 are_congruent_mod 623 &
          32*32*32*32*2|^3 = 623*13464+536 by Lem1,Lm3;
        then 2|^(38*4+3),536 are_congruent_mod 623 &
          536*536*536*536*2|^3 = 623*1059890743+39 by LemS;
        then 2|^(155*4+3),39 are_congruent_mod 623 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 624;
        2|^9 = 624*0+512 &
          512*512*512*512*2|^3 = 624*881018932+320 by Lm9,Lm3;
        then 2|^(9*4+3),320 are_congruent_mod 624 &
          320*320*320*320*2|^0 = 624*16804102+352 by Lem1,Lm1;
        then 2|^(39*4+0),352 are_congruent_mod 624 &
          352*352*352*352*2|^0 = 624*24602886+352 by LemS;
        then 2|^(156*4+0),352 are_congruent_mod 624 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 625;
        2|^9 = 625*0+512 &
          512*512*512*512*2|^3 = 625*879609302+138 by Lm9,Lm3;
        then 2|^(9*4+3),138 are_congruent_mod 625 &
          138*138*138*138*2|^0 = 625*580278+186 by Lem1,Lm1;
        then 2|^(39*4+0),186 are_congruent_mod 625 &
          186*186*186*186*2|^1 = 625*3830026+182 by LemS;
        then 2|^(156*4+1),182 are_congruent_mod 625 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 626;
        2|^9 = 626*0+512 &
          512*512*512*512*2|^3 = 626*878204175+338 by Lm9,Lm3;
        then 2|^(9*4+3),338 are_congruent_mod 626 &
          338*338*338*338*2|^0 = 626*20849347+314 by Lem1,Lm1;
        then 2|^(39*4+0),314 are_congruent_mod 626 &
          314*314*314*314*2|^2 = 626*62116110+4 by LemS,Lm2;
        then 2|^(156*4+2),4 are_congruent_mod 626 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 627;
        2|^9 = 627*0+512 &
          512*512*512*512*2|^3 = 627*876803530+578 by Lm9,Lm3;
        then 2|^(9*4+3),578 are_congruent_mod 627 &
          578*578*578*578*2|^0 = 627*178009759+163 by Lem1,Lm1;
        then 2|^(39*4+0),163 are_congruent_mod 627 &
          163*163*163*163*2|^3 = 627*9006848+392 by LemS,Lm3;
        then 2|^(156*4+3),392 are_congruent_mod 627 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 628;
        2|^9 = 628*0+512 &
          512*512*512*512*2|^3 = 628*875407346+600 by Lm9,Lm3;
        then 2|^(9*4+3),600 are_congruent_mod 628 &
          600*600*600*600*2|^1 = 628*412738853+316 by Lem1;
        then 2|^(39*4+1),316 are_congruent_mod 628 &
          316*316*316*316*2|^0 = 628*15877740+16 by LemS,Lm1;
        then 2|^(157*4+0),16 are_congruent_mod 628 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 629;
        2|^9 = 629*0+512 &
          512*512*512*512*2|^3 = 629*874015602+230 by Lm9,Lm3;
        then 2|^(9*4+3),230 are_congruent_mod 629 &
          230*230*230*230*2|^1 = 629*8897965+15 by Lem1;
        then 2|^(39*4+1),15 are_congruent_mod 629 &
          15*15*15*15*2|^1 = 629*160+610 by LemS;
        then 2|^(157*4+1),610 are_congruent_mod 629 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 630;
        2|^9 = 630*0+512 & 512*512*512*512*2|^3 = 630*872628276+8 by Lm9,Lm3;
        then 2|^(9*4+3),8 are_congruent_mod 630 &
          8*8*8*8*2|^1 = 630*13+2 by Lem1;
        then 2|^(39*4+1),2 are_congruent_mod 630 &
          2*2*2*2*2|^2 = 630*0+64 by LemS,Lm2;
        then 2|^(157*4+2),64 are_congruent_mod 630 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 631;
        hence thesis by XPRIMES1:631;
      end;
      suppose A: n = 632;
        2|^9 = 632*0+512 & 512*512*512*512*2|^3 = 632*869866794+80 by Lm9,Lm3;
        then 2|^(9*4+3),80 are_congruent_mod 632 &
          80*80*80*80*2|^2 = 632*259240+320 by Lem1,Lm2;
        then 2|^(39*4+2),320 are_congruent_mod 632 &
          320*320*320*320*2|^0 = 632*16591392+256 by LemS,Lm1;
        then 2|^(158*4+0),256 are_congruent_mod 632 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 633;
        2|^9 = 633*0+512 &
          512*512*512*512*2|^3 = 633*868492596+620 by Lm9,Lm3;
        then 2|^(9*4+3),620 are_congruent_mod 633 &
          620*620*620*620*2|^2 = 633*933733712+304 by Lem1,Lm2;
        then 2|^(39*4+2),304 are_congruent_mod 633 &
          304*304*304*304*2|^1 = 633*26984888+8 by LemS;
        then 2|^(158*4+1),8 are_congruent_mod 633 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 634;
        2|^9 = 634*0+512 &
          512*512*512*512*2|^3 = 634*867122734+532 by Lm9,Lm3;
        then 2|^(9*4+3),532 are_congruent_mod 634 &
          532*532*532*532*2|^2 = 634*505379082+316 by Lem1,Lm2;
        then 2|^(39*4+2),316 are_congruent_mod 634 &
          316*316*316*316*2|^2 = 634*62909910+4 by LemS;
        then 2|^(158*4+2),4 are_congruent_mod 634 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 635;
        2|^9 = 635*0+512 &
          512*512*512*512*2|^3 = 635*865757187+143 by Lm9,Lm3;
        then 2|^(9*4+3),143 are_congruent_mod 635 &
          143*143*143*143*2|^2 = 635*2634088+524 by Lem1,Lm2;
        then 2|^(39*4+2),524 are_congruent_mod 635 &
          524*524*524*524*2|^3 = 635*949820217+413 by LemS,Lm3;
        then 2|^(158*4+3),413 are_congruent_mod 635 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 636;
        2|^9 = 636*0+512 &
          512*512*512*512*2|^3 = 636*864395933+500 by Lm9,Lm3;
        then 2|^(9*4+3),500 are_congruent_mod 636 &
          500*500*500*500*2|^3 = 636*786163522+8 by Lem1;
        then 2|^(39*4+3),8 are_congruent_mod 636 &
          8*8*8*8*2|^0 = 636*6+280 by LemS,Lm1;
        then 2|^(159*4+0),280 are_congruent_mod 636 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 637;
        2|^9 = 637*0+512 &
          512*512*512*512*2|^3 = 637*863038954+190 by Lm9,Lm3;
        then 2|^(9*4+3),190 are_congruent_mod 637 &
          190*190*190*190*2|^3 = 637*16366844+372 by Lem1;
        then 2|^(39*4+3),372 are_congruent_mod 637 &
          372*372*372*372*2|^1 = 637*60126001+275 by LemS;
        then 2|^(159*4+1),275 are_congruent_mod 637 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 638;
        2|^9 = 638*0+512 &
          512*512*512*512*2|^3 = 638*861686228+424 by Lm9,Lm3;
        then 2|^(9*4+3),424 are_congruent_mod 638 &
          424*424*424*424*2|^3 = 638*405259061+490 by Lem1;
        then 2|^(39*4+3),490 are_congruent_mod 638 &
          490*490*490*490*2|^2 = 638*361429529+498 by LemS,Lm2;
        then 2|^(159*4+2),498 are_congruent_mod 638 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 639;
        2|^9 = 639*0+512 &
          512*512*512*512*2|^3 = 639*860337736+584 by Lm9,Lm3;
        then 2|^(9*4+3),584 are_congruent_mod 639 &
          584*584*584*584*2|^3 = 639*1456265353+521 by Lem1;
        then 2|^(39*4+3),521 are_congruent_mod 639 &
          521*521*521*521*2|^3 = 639*922444024+512 by LemS;
        then 2|^(159*4+3),512 are_congruent_mod 639 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 640;
        2|^10 = 640*1+384 &
          384*384*384*384*2|^0 = 640*33973862+256 by Lm10,Lm1;
        then 2|^(10*4+0),256 are_congruent_mod 640 &
          256*256*256*256*2|^0 = 640*6710886+256 by Lem1;
        then 2|^(40*4+0),256 are_congruent_mod 640 &
          256*256*256*256*2|^0 = 640*6710886+256 by LemS;
        then 2|^(160*4+0),256 are_congruent_mod 640 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 641;
        hence thesis by XPRIMES1:641;
      end;
      suppose A: n = 642;
        2|^10 = 642*1+382 &
          382*382*382*382*2|^0 = 642*33167934+148 by Lm10,Lm1;
        then 2|^(10*4+0),148 are_congruent_mod 642 &
          148*148*148*148*2|^0 = 642*747328+640 by Lem1;
        then 2|^(40*4+0),640 are_congruent_mod 642 &
          640*640*640*640*2|^2 = 642*1045309408+64 by LemS,Lm2;
        then 2|^(160*4+2),64 are_congruent_mod 642 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 643;
        hence thesis by XPRIMES1:643;
      end;
      suppose A: n = 644;
        2|^10 = 644*1+380 &
          380*380*380*380*2|^0 = 644*32377888+128 by Lm10,Lm1;
        then 2|^(10*4+0),128 are_congruent_mod 644 &
          128*128*128*128*2|^1 = 644*833650+312 by Lem1;
        then 2|^(40*4+1),312 are_congruent_mod 644 &
          312*312*312*312*2|^0 = 644*14714059+340 by LemS,Lm1;
        then 2|^(161*4+0),340 are_congruent_mod 644 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 645;
        hence thesis by A5,ENUMSET1:def 2;
      end;
      suppose A: n = 646;
        2|^10 = 646*1+378 &
          378*378*378*378*2|^0 = 646*31603463+358 by Lm10,Lm1;
        then 2|^(10*4+0),358 are_congruent_mod 646 &
          358*358*358*358*2|^1 = 646*50854522+580 by Lem1;
        then 2|^(40*4+1),580 are_congruent_mod 646 &
          580*580*580*580*2|^2 = 646*700711826+404 by LemS,Lm2;
        then 2|^(161*4+2),404 are_congruent_mod 646 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 647;
        hence thesis by XPRIMES1:647;
      end;
      suppose A: n = 648;
        2|^10 = 648*1+376 &
          376*376*376*376*2|^0 = 648*30844403+232 by Lm10,Lm1;
        then 2|^(10*4+0),232 are_congruent_mod 648 &
          232*232*232*232*2|^2 = 648*17882857+568 by Lem1,Lm2;
        then 2|^(40*4+2),568 are_congruent_mod 648 &
          568*568*568*568*2|^0 = 648*160626921+568 by LemS,Lm1;
        then 2|^(162*4+0),568 are_congruent_mod 648 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 649;
        2|^10 = 649*1+375 &
          375*375*375*375*2|^0 = 649*30470555+430 by Lm10,Lm1;
        then 2|^(10*4+0),430 are_congruent_mod 649 &
          430*430*430*430*2|^2 = 649*210711926+26 by Lem1,Lm2;
        then 2|^(40*4+2),26 are_congruent_mod 649 &
          26*26*26*26*2|^1 = 649*1408+160 by LemS;
        then 2|^(162*4+1),160 are_congruent_mod 649 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 650;
        2|^10 = 650*1+374 &
          374*374*374*374*2|^0 = 650*30100454+276 by Lm10,Lm1;
        then 2|^(10*4+0),276 are_congruent_mod 650 &
          276*276*276*276*2|^2 = 650*35709433+454 by Lem1,Lm2;
        then 2|^(40*4+2),454 are_congruent_mod 650 &
          454*454*454*454*2|^2 = 650*261438802+524 by LemS;
        then 2|^(162*4+2),524 are_congruent_mod 650 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 651;
        2|^10 = 651*1+373 &
          373*373*373*373*2|^0 = 651*29734068+373 by Lm10,Lm1;
        then 2|^(10*4+0),373 are_congruent_mod 651 &
          373*373*373*373*2|^2 = 651*118936274+190 by Lem1,Lm2;
        then 2|^(40*4+2),190 are_congruent_mod 651 &
          190*190*190*190*2|^3 = 651*16014869+281 by LemS,Lm3;
        then 2|^(162*4+3),281 are_congruent_mod 651 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 652;
        2|^10 = 652*1+372 &
          372*372*372*372*2|^0 = 652*29371367+172 by Lm10,Lm1;
        then 2|^(10*4+0),172 are_congruent_mod 652 &
          172*172*172*172*2|^3 = 652*10738810+328 by Lem1,Lm3;
        then 2|^(40*4+3),328 are_congruent_mod 652 &
          328*328*328*328*2|^0 = 652*17752020+16 by LemS,Lm1;
        then 2|^(163*4+0),16 are_congruent_mod 652 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 653;
        hence thesis by XPRIMES1:653;
      end;
      suppose A: n = 654;
        2|^10 = 654*1+370 &
          370*370*370*370*2|^0 = 654*28656896+16 by Lm10,Lm1;
        then 2|^(10*4+0),16 are_congruent_mod 654 &
          16*16*16*16*2|^3 = 654*801+434 by Lem1,Lm3;
        then 2|^(40*4+3),434 are_congruent_mod 654 &
          434*434*434*434*2|^2 = 654*216990720+64 by LemS,Lm2;
        then 2|^(163*4+2),64 are_congruent_mod 654 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 655;
        2|^10 = 655*1+369 &
          369*369*369*369*2|^0 = 655*28305065+346 by Lm10,Lm1;
        then 2|^(10*4+0),346 are_congruent_mod 655 &
          346*346*346*346*2|^3 = 655*175046359+103 by Lem1,Lm3;
        then 2|^(40*4+3),103 are_congruent_mod 655 &
          103*103*103*103*2|^3 = 655*1374667+163 by LemS;
        then 2|^(163*4+3),163 are_congruent_mod 655 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 656;
        2|^10 = 656*1+368 &
          368*368*368*368*2|^1 = 656*55913596+576 by Lm10;
        then 2|^(10*4+1),576 are_congruent_mod 656 &
          576*576*576*576*2|^0 = 656*167797735+16 by Lem1,Lm1;
        then 2|^(41*4+0),16 are_congruent_mod 656 &
          16*16*16*16*2|^0 = 656*99+592 by LemS;
        then 2|^(164*4+0),592 are_congruent_mod 656 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 657;
        2|^10 = 657*1+367 &
          367*367*367*367*2|^1 = 657*55224130+32 by Lm10;
        then 2|^(10*4+1),32 are_congruent_mod 657 &
          32*32*32*32*2|^0 = 657*1596+4 by Lem1,Lm1;
        then 2|^(41*4+0),4 are_congruent_mod 657 &
          4*4*4*4*2|^1 = 657*0+512 by LemS;
        then 2|^(164*4+1),512 are_congruent_mod 657 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 658;
        2|^10 = 658*1+366 &
          366*366*366*366*2|^1 = 658*54541671+354 by Lm10;
        then 2|^(10*4+1),354 are_congruent_mod 658 &
          354*354*354*354*2|^0 = 658*23866413+102 by Lem1,Lm1;
        then 2|^(41*4+0),102 are_congruent_mod 658 &
          102*102*102*102*2|^2 = 658*658013+310 by LemS,Lm2;
        then 2|^(164*4+2),310 are_congruent_mod 658 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 659;
        hence thesis by XPRIMES1:659;
      end;
      suppose A: n = 660;
        2|^10 = 660*1+364 &
          364*364*364*364*2|^1 = 660*53197545+332 by Lm10;
        then 2|^(10*4+1),332 are_congruent_mod 660 &
          332*332*332*332*2|^1 = 660*36816152+32 by Lem1;
        then 2|^(41*4+1),32 are_congruent_mod 660 &
          32*32*32*32*2|^0 = 660*1588+496 by LemS,Lm1;
        then 2|^(165*4+0),496 are_congruent_mod 660 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 661;
        hence thesis by XPRIMES1:661;
      end;
      suppose A: n = 662;
        2|^10 = 662*1+362 &
          362*362*362*362*2|^1 = 662*51880755+62 by Lm10;
        then 2|^(10*4+1),62 are_congruent_mod 662 &
          62*62*62*62*2|^1 = 662*44641+330 by Lem1;
        then 2|^(41*4+1),330 are_congruent_mod 662 &
          330*330*330*330*2|^2 = 662*71656858+4 by LemS,Lm2;
        then 2|^(165*4+2),4 are_congruent_mod 662 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 663;
        2|^10 = 663*1+361 &
          361*361*361*361*2|^1 = 663*51232467+461 by Lm10;
        then 2|^(10*4+1),461 are_congruent_mod 663 &
          461*461*461*461*2|^1 = 663*136244873+83 by Lem1;
        then 2|^(41*4+1),83 are_congruent_mod 663 &
          83*83*83*83*2|^3 = 663*572649+281 by LemS,Lm3;
        then 2|^(165*4+3),281 are_congruent_mod 663 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 664;
        2|^10 = 664*1+360 &
          360*360*360*360*2|^1 = 664*50590843+248 by Lm10;
        then 2|^(10*4+1),248 are_congruent_mod 664 &
          248*248*248*248*2|^2 = 664*22787602+336 by Lem1,Lm2;
        then 2|^(41*4+2),336 are_congruent_mod 664 &
          336*336*336*336*2|^0 = 664*19195040+256 by LemS,Lm1;
        then 2|^(166*4+0),256 are_congruent_mod 664 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 665;
        2|^10 = 665*1+359 &
          359*359*359*359*2|^1 = 665*49955826+32 by Lm10;
        then 2|^(10*4+1),32 are_congruent_mod 665 &
          32*32*32*32*2|^2 = 665*6307+149 by Lem1,Lm2;
        then 2|^(41*4+2),149 are_congruent_mod 665 &
          149*149*149*149*2|^1 = 665*1482359+67 by LemS;
        then 2|^(166*4+1),67 are_congruent_mod 665 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 666;
        2|^10 = 666*1+358 &
          358*358*358*358*2|^1 = 666*49327360+32 by Lm10;
        then 2|^(10*4+1),32 are_congruent_mod 666 &
          32*32*32*32*2|^2 = 666*6297+502 by Lem1,Lm2;
        then 2|^(41*4+2),502 are_congruent_mod 666 &
          502*502*502*502*2|^2 = 666*381417513+406 by LemS;
        then 2|^(166*4+2),406 are_congruent_mod 666 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 667;
        2|^10 = 667*1+357 &
          357*357*357*357*2|^1 = 667*48705390+72 by Lm10;
        then 2|^(10*4+1),72 are_congruent_mod 667 &
          72*72*72*72*2|^2 = 667*161162+370 by Lem1,Lm2;
        then 2|^(41*4+2),370 are_congruent_mod 667 &
          370*370*370*370*2|^3 = 667*224786926+358 by LemS,Lm3;
        then 2|^(166*4+3),358 are_congruent_mod 667 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 668;
        2|^10 = 668*1+356 &
          356*356*356*356*2|^1 = 668*48089861+244 by Lm10;
        then 2|^(10*4+1),244 are_congruent_mod 668 &
          244*244*244*244*2|^3 = 668*42449524+336 by Lem1,Lm3;
        then 2|^(41*4+3),336 are_congruent_mod 668 &
          336*336*336*336*2|^0 = 668*19080100+16 by LemS,Lm1;
        then 2|^(167*4+0),16 are_congruent_mod 668 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 669;
        2|^10 = 669*1+355 &
          355*355*355*355*2|^1 = 669*47480719+239 by Lm10;
        then 2|^(10*4+1),239 are_congruent_mod 669 &
          239*239*239*239*2|^3 = 669*39017143+461 by Lem1,Lm3;
        then 2|^(41*4+3),461 are_congruent_mod 669 &
          461*461*461*461*2|^1 = 669*135022946+8 by LemS;
        then 2|^(167*4+1),8 are_congruent_mod 669 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 670;
        2|^10 = 670*1+354 &
          354*354*354*354*2|^1 = 670*46877910+12 by Lm10;
        then 2|^(10*4+1),12 are_congruent_mod 670 &
          12*12*12*12*2|^3 = 670*247+398 by Lem1,Lm3;
        then 2|^(41*4+3),398 are_congruent_mod 670 &
          398*398*398*398*2|^2 = 670*149801953+354 by LemS,Lm2;
        then 2|^(167*4+2),354 are_congruent_mod 670 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 671;
        2|^10 = 671*1+353 &
          353*353*353*353*2|^1 = 671*46281379+453 by Lm10;
        then 2|^(10*4+1),453 are_congruent_mod 671 &
          453*453*453*453*2|^3 = 671*502065379+139 by Lem1,Lm3;
        then 2|^(41*4+3),139 are_congruent_mod 671 &
          139*139*139*139*2|^3 = 671*4450683+35 by LemS;
        then 2|^(167*4+3),35 are_congruent_mod 671 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 672;
        2|^10 = 672*1+352 &
          352*352*352*352*2|^2 = 672*91382150+64 by Lm10,Lm2;
        then 2|^(10*4+2),64 are_congruent_mod 672 &
          64*64*64*64*2|^0 = 672*24966+64 by Lem1,Lm1;
        then 2|^(42*4+0),64 are_congruent_mod 672 &
          64*64*64*64*2|^0 = 672*24966+64 by LemS;
        then 2|^(168*4+0),64 are_congruent_mod 672 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 673;
        hence thesis by XPRIMES1:673;
      end;
      suppose A: n = 674;
        2|^10 = 674*1+350 &
          350*350*350*350*2|^2 = 674*89057863+338 by Lm10,Lm2;
        then 2|^(10*4+2),338 are_congruent_mod 674 &
          338*338*338*338*2|^0 = 674*19364527+338 by Lem1,Lm1;
        then 2|^(42*4+0),338 are_congruent_mod 674 &
          338*338*338*338*2|^2 = 674*77458110+4 by LemS,Lm2;
        then 2|^(168*4+2),4 are_congruent_mod 674 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 675;
        2|^10 = 675*1+349 &
          349*349*349*349*2|^2 = 675*87913976+604 by Lm10,Lm2;
        then 2|^(10*4+2),604 are_congruent_mod 675 &
          604*604*604*604*2|^0 = 675*197171427+631 by Lem1,Lm1;
        then 2|^(42*4+0),631 are_congruent_mod 675 &
          631*631*631*631*2|^3 = 675*1878899933+593 by LemS,Lm3;
        then 2|^(168*4+3),593 are_congruent_mod 675 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 676;
        2|^10 = 676*1+348 &
          348*348*348*348*2|^2 = 676*86782123+116 by Lm10,Lm2;
        then 2|^(10*4+2),116 are_congruent_mod 676 &
          116*116*116*116*2|^1 = 676*535692+80 by Lem1;
        then 2|^(42*4+1),80 are_congruent_mod 676 &
          80*80*80*80*2|^0 = 676*60591+484 by LemS,Lm1;
        then 2|^(169*4+0),484 are_congruent_mod 676 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 677;
        hence thesis by XPRIMES1:677;
      end;
      suppose A: n = 678;
        2|^10 = 678*1+346 &
          346*346*346*346*2|^2 = 678*84554104+112 by Lm10,Lm2;
        then 2|^(10*4+2),112 are_congruent_mod 678 &
          112*112*112*112*2|^1 = 678*464165+2 by Lem1;
        then 2|^(42*4+1),2 are_congruent_mod 678 &
          2*2*2*2*2|^2 = 678*0+64 by LemS,Lm2;
        then 2|^(169*4+2),64 are_congruent_mod 678 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 679;
        2|^10 = 679*1+345 &
          345*345*345*345*2|^2 = 679*83457735+435 by Lm10,Lm2;
        then 2|^(10*4+2),435 are_congruent_mod 679 &
          435*435*435*435*2|^1 = 679*105467159+289 by Lem1;
        then 2|^(42*4+1),289 are_congruent_mod 679 &
          289*289*289*289*2|^3 = 679*82188600+128 by LemS,Lm3;
        then 2|^(169*4+3),128 are_congruent_mod 679 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 680;
        2|^10 = 680*1+344 &
          344*344*344*344*2|^2 = 680*82372993+344 by Lm10,Lm2;
        then 2|^(10*4+2),344 are_congruent_mod 680 &
          344*344*344*344*2|^2 = 680*82372993+344 by Lem1;
        then 2|^(42*4+2),344 are_congruent_mod 680 &
          344*344*344*344*2|^0 = 680*20593248+256 by LemS,Lm1;
        then 2|^(170*4+0),256 are_congruent_mod 680 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 681;
        2|^10 = 681*1+343 &
          343*343*343*343*2|^2 = 681*81299777+667 by Lm10,Lm2;
        then 2|^(10*4+2),667 are_congruent_mod 681 &
          667*667*667*667*2|^2 = 681*1162562245+439 by Lem1;
        then 2|^(42*4+2),439 are_congruent_mod 681 &
          439*439*439*439*2|^1 = 681*109078954+8 by LemS;
        then 2|^(170*4+1),8 are_congruent_mod 681 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 682;
        2|^10 = 682*1+342 & 342*342*342*342*2|^2 = 682*80237990+4 by Lm10,Lm2;
        then 2|^(10*4+2),4 are_congruent_mod 682 &
          4*4*4*4*2|^2 = 682*1+342 by Lem1;
        then 2|^(42*4+2),342 are_congruent_mod 682 &
          342*342*342*342*2|^2 = 682*80237990+4 by LemS;
        then 2|^(170*4+2),4 are_congruent_mod 682 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 683;
        hence thesis by XPRIMES1:683;
      end;
      suppose A: n = 684;
        2|^10 = 684*1+340 &
          340*340*340*340*2|^2 = 684*78148304+64 by Lm10,Lm2;
        then 2|^(10*4+2),64 are_congruent_mod 684 &
          64*64*64*64*2|^3 = 684*196224+512 by Lem1,Lm3;
        then 2|^(42*4+3),512 are_congruent_mod 684 &
          512*512*512*512*2|^0 = 684*100467071+172 by LemS,Lm1;
        then 2|^(171*4+0),172 are_congruent_mod 684 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 685;
        2|^10 = 685*1+339 &
          339*339*339*339*2|^2 = 685*77120211+429 by Lm10,Lm2;
        then 2|^(10*4+2),429 are_congruent_mod 685 &
          429*429*429*429*2|^3 = 685*395574769+683 by Lem1,Lm3;
        then 2|^(42*4+3),683 are_congruent_mod 685 &
          683*683*683*683*2|^1 = 685*635363466+32 by LemS;
        then 2|^(171*4+1),32 are_congruent_mod 685 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 686;
        2|^10 = 686*1+338 &
          338*338*338*338*2|^2 = 686*76103157+442 by Lm10,Lm2;
        then 2|^(10*4+2),442 are_congruent_mod 686 &
          442*442*442*442*2|^3 = 686*445097288+400 by Lem1,Lm3;
        then 2|^(42*4+3),400 are_congruent_mod 686 &
          400*400*400*400*2|^2 = 686*149271137+18 by LemS,Lm2;
        then 2|^(171*4+2),18 are_congruent_mod 686 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 687;
        2|^10 = 687*1+337 &
          337*337*337*337*2|^2 = 687*75097046+442 by Lm10,Lm2;
        then 2|^(10*4+2),442 are_congruent_mod 687 &
          442*442*442*442*2|^3 = 687*444449403+107 by Lem1,Lm3;
        then 2|^(42*4+3),107 are_congruent_mod 687 &
          107*107*107*107*2|^3 = 687*1526400+8 by LemS;
        then 2|^(171*4+3),8 are_congruent_mod 687 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 688;
        2|^10 = 688*1+336 &
          336*336*336*336*2|^3 = 688*148203567+432 by Lm10,Lm3;
        then 2|^(10*4+3),432 are_congruent_mod 688 &
          432*432*432*432*2|^0 = 688*50622845+16 by Lem1,Lm1;
        then 2|^(43*4+0),16 are_congruent_mod 688 &
          16*16*16*16*2|^0 = 688*95+176 by LemS;
        then 2|^(172*4+0),176 are_congruent_mod 688 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 689;
        2|^10 = 689*1+335 &
          335*335*335*335*2|^3 = 689*146234550+50 by Lm10,Lm3;
        then 2|^(10*4+3),50 are_congruent_mod 689 &
          50*50*50*50*2|^0 = 689*9071+81 by Lem1,Lm1;
        then 2|^(43*4+0),81 are_congruent_mod 689 &
          81*81*81*81*2|^1 = 689*124954+136 by LemS;
        then 2|^(172*4+1),136 are_congruent_mod 689 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 690;
        2|^10 = 690*1+334 &
          334*334*334*334*2|^3 = 690*144286853+518 by Lm10,Lm3;
        then 2|^(10*4+3),518 are_congruent_mod 690 &
          518*518*518*518*2|^0 = 690*104344592+496 by Lem1,Lm1;
        then 2|^(43*4+0),496 are_congruent_mod 690 &
          496*496*496*496*2|^2 = 690*350863027+394 by LemS,Lm2;
        then 2|^(172*4+2),394 are_congruent_mod 690 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 691;
        hence thesis by XPRIMES1:691;
      end;
      suppose A: n = 692;
        2|^10 = 692*1+332 &
          332*332*332*332*2|^3 = 692*140454684+80 by Lm10,Lm3;
        then 2|^(10*4+3),80 are_congruent_mod 692 &
          80*80*80*80*2|^1 = 692*118381+348 by Lem1;
        then 2|^(43*4+1),348 are_congruent_mod 692 &
          348*348*348*348*2|^0 = 692*21193900+16 by LemS,Lm1;
        then 2|^(173*4+0),16 are_congruent_mod 692 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 693;
        2|^10 = 693*1+331 &
          331*331*331*331*2|^3 = 693*138569843+569 by Lm10,Lm3;
        then 2|^(10*4+3),569 are_congruent_mod 693 &
          569*569*569*569*2|^1 = 693*302514242+536 by Lem1;
        then 2|^(43*4+1),536 are_congruent_mod 693 &
          536*536*536*536*2|^1 = 693*238207768+8 by LemS;
        then 2|^(173*4+1),8 are_congruent_mod 693 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 694;
        2|^10 = 694*1+330 &
          330*330*330*330*2|^3 = 694*136705590+540 by Lm10,Lm3;
        then 2|^(10*4+3),540 are_congruent_mod 694 &
          540*540*540*540*2|^1 = 694*245044841+346 by Lem1;
        then 2|^(43*4+1),346 are_congruent_mod 694 &
          346*346*346*346*2|^2 = 694*82604730+4 by LemS,Lm2;
        then 2|^(173*4+2),4 are_congruent_mod 694 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 695;
        2|^10 = 695*1+329 &
          329*329*329*329*2|^3 = 695*134861744+568 by Lm10,Lm3;
        then 2|^(10*4+3),568 are_congruent_mod 695 &
          568*568*568*568*2|^1 = 695*299528763+467 by Lem1;
        then 2|^(43*4+1),467 are_congruent_mod 695 &
          467*467*467*467*2|^3 = 695*547485604+588 by LemS,Lm3;
        then 2|^(173*4+3),588 are_congruent_mod 695 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 696;
        2|^10 = 696*1+328 &
          328*328*328*328*2|^3 = 696*133038127+56 by Lm10,Lm3;
        then 2|^(10*4+3),56 are_congruent_mod 696 &
          56*56*56*56*2|^2 = 696*56520+64 by Lem1,Lm2;
        then 2|^(43*4+2),64 are_congruent_mod 696 &
          64*64*64*64*2|^0 = 696*24105+136 by LemS,Lm1;
        then 2|^(174*4+0),136 are_congruent_mod 696 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 697;
        2|^10 = 697*1+327 &
          327*327*327*327*2|^3 = 697*131234560+8 by Lm10,Lm3;
        then 2|^(10*4+3),8 are_congruent_mod 697 &
          8*8*8*8*2|^2 = 697*23+353 by Lem1,Lm2;
        then 2|^(43*4+2),353 are_congruent_mod 697 &
          353*353*353*353*2|^1 = 697*44554958+36 by LemS;
        then 2|^(174*4+1),36 are_congruent_mod 697 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 698;
        2|^10 = 698*1+326 &
          326*326*326*326*2|^3 = 698*129450867+242 by Lm10,Lm3;
        then 2|^(10*4+3),242 are_congruent_mod 698 &
          242*242*242*242*2|^2 = 698*19654682+348 by Lem1,Lm2;
        then 2|^(43*4+2),348 are_congruent_mod 698 &
          348*348*348*348*2|^2 = 698*84046870+4 by LemS;
        then 2|^(174*4+2),4 are_congruent_mod 698 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 699;
        2|^10 = 699*1+325 &
          325*325*325*325*2|^3 = 699*127686874+74 by Lm10,Lm3;
        then 2|^(10*4+3),74 are_congruent_mod 699 &
          74*74*74*74*2|^2 = 699*171597+1 by Lem1,Lm2;
        then 2|^(43*4+2),1 are_congruent_mod 699 &
          1*1*1*1*2|^3 = 699*0+8 by LemS,Lm3;
        then 2|^(174*4+3),8 are_congruent_mod 699 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 700;
        2|^10 = 700*1+324 &
          324*324*324*324*2|^3 = 700*125942406+408 by Lm10,Lm3;
        then 2|^(10*4+3),408 are_congruent_mod 700 &
          408*408*408*408*2|^3 = 700*316688723+268 by Lem1;
        then 2|^(43*4+3),268 are_congruent_mod 700 &
          268*268*268*268*2|^0 = 700*7369552+576 by LemS,Lm1;
        then 2|^(175*4+0),576 are_congruent_mod 700 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 701;
        hence thesis by XPRIMES1:701;
      end;
      suppose A: n = 702;
        2|^10 = 702*1+322 &
          322*322*322*322*2|^3 = 702*122511360+128 by Lm10,Lm3;
        then 2|^(10*4+3),128 are_congruent_mod 702 &
          128*128*128*128*2|^3 = 702*3059093+362 by Lem1;
        then 2|^(43*4+3),362 are_congruent_mod 702 &
          362*362*362*362*2|^2 = 702*97849173+298 by LemS,Lm2;
        then 2|^(175*4+2),298 are_congruent_mod 702 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 703;
        2|^10 = 703*1+321 &
          321*321*321*321*2|^3 = 703*120824440+128 by Lm10,Lm3;
        then 2|^(10*4+3),128 are_congruent_mod 703 &
          128*128*128*128*2|^3 = 703*3054742+22 by Lem1;
        then 2|^(43*4+3),22 are_congruent_mod 703 &
          22*22*22*22*2|^3 = 703*2665+553 by LemS;
        then 2|^(175*4+3),553 are_congruent_mod 703 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 704;
        2|^11 = 704*2+640 &
          640*640*640*640*2|^0 = 704*238312727+192 by Lm11,Lm1;
        then 2|^(11*4+0),192 are_congruent_mod 704 &
          192*192*192*192*2|^0 = 704*1930333+64 by Lem1;
        then 2|^(44*4+0),64 are_congruent_mod 704 &
          64*64*64*64*2|^0 = 704*23831+192 by LemS;
        then 2|^(176*4+0),192 are_congruent_mod 704 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 704 < n & n <= 804;
      then 704+1 <= n & n <= 804 by NAT_1:13;
      then n = 705 or ... or n = 804;
      then per cases;
      suppose A: n = 705;
        2|^11 = 705*2+638 &
          638*638*638*638*2|^0 = 705*235013926+106 by Lm11,Lm1;
        then 2|^(11*4+0),106 are_congruent_mod 705 &
          106*106*106*106*2|^0 = 705*179074+526 by Lem1;
        then 2|^(44*4+0),526 are_congruent_mod 705 &
          526*526*526*526*2|^1 = 705*217162011+197 by LemS;
        then 2|^(176*4+1),197 are_congruent_mod 705 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 706;
        2|^11 = 706*2+636 &
          636*636*636*636*2|^0 = 706*231752144+352 by Lm11,Lm1;
        then 2|^(11*4+0),352 are_congruent_mod 706 &
          352*352*352*352*2|^0 = 706*21745327+354 by Lem1;
        then 2|^(44*4+0),354 are_congruent_mod 706 &
          354*354*354*354*2|^2 = 706*88975070+4 by LemS,Lm2;
        then 2|^(176*4+2),4 are_congruent_mod 706 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 707;
        2|^11 = 707*2+634 &
          634*634*634*634*2|^0 = 707*228527052+172 by Lm11,Lm1;
        then 2|^(11*4+0),172 are_congruent_mod 707 &
          172*172*172*172*2|^0 = 707*1237925+81 by Lem1;
        then 2|^(44*4+0),81 are_congruent_mod 707 &
          81*81*81*81*2|^3 = 707*487091+431 by LemS,Lm3;
        then 2|^(176*4+3),431 are_congruent_mod 707 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 708;
        2|^11 = 708*2+632 &
          632*632*632*632*2|^0 = 708*225338321+508 by Lm11,Lm1;
        then 2|^(11*4+0),508 are_congruent_mod 708 &
          508*508*508*508*2|^1 = 708*188127198+8 by Lem1;
        then 2|^(44*4+1),8 are_congruent_mod 708 &
          8*8*8*8*2|^0 = 708*5+556 by LemS,Lm1;
        then 2|^(177*4+0),556 are_congruent_mod 708 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 709;
        hence thesis by XPRIMES1:709;
      end;
      suppose A: n = 710;
        2|^11 = 710*2+628 &
          628*628*628*628*2|^0 = 710*219068647+86 by Lm11,Lm1;
        then 2|^(11*4+0),86 are_congruent_mod 710 &
          86*86*86*86*2|^1 = 710*154086+572 by Lem1;
        then 2|^(44*4+1),572 are_congruent_mod 710 &
          572*572*572*572*2|^2 = 710*603095041+314 by LemS,Lm2;
        then 2|^(177*4+2),314 are_congruent_mod 710 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 711;
        2|^11 = 711*2+626 &
          626*626*626*626*2|^0 = 711*215987059+427 by Lm11,Lm1;
        then 2|^(11*4+0),427 are_congruent_mod 711 &
          427*427*427*427*2|^1 = 711*93512979+413 by Lem1;
        then 2|^(44*4+1),413 are_congruent_mod 711 &
          413*413*413*413*2|^3 = 711*327356216+512 by LemS,Lm3;
        then 2|^(177*4+3),512 are_congruent_mod 711 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 712;
        2|^11 = 712*2+624 &
          624*624*624*624*2|^0 = 712*212940546+624 by Lm11,Lm1;
        then 2|^(11*4+0),624 are_congruent_mod 712 &
          624*624*624*624*2|^2 = 712*851762187+360 by Lem1,Lm2;
        then 2|^(44*4+2),360 are_congruent_mod 712 &
          360*360*360*360*2|^0 = 712*23590112+256 by LemS,Lm1;
        then 2|^(178*4+0),256 are_congruent_mod 712 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 713;
        2|^11 = 713*2+622 &
          622*622*622*622*2|^0 = 713*209928793+47 by Lm11,Lm1;
        then 2|^(11*4+0),47 are_congruent_mod 713 &
          47*47*47*47*2|^2 = 713*27375+349 by Lem1,Lm2;
        then 2|^(44*4+2),349 are_congruent_mod 713 &
          349*349*349*349*2|^1 = 713*41614259+535 by LemS;
        then 2|^(178*4+1),535 are_congruent_mod 713 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 714;
        2|^11 = 714*2+620 &
          620*620*620*620*2|^0 = 714*206951484+424 by Lm11,Lm1;
        then 2|^(11*4+0),424 are_congruent_mod 714 &
          424*424*424*424*2|^2 = 714*181061121+310 by Lem1,Lm2;
        then 2|^(44*4+2),310 are_congruent_mod 714 &
          310*310*310*310*2|^2 = 714*51737871+106 by LemS;
        then 2|^(178*4+2),106 are_congruent_mod 714 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 715;
        2|^11 = 715*2+618 &
          618*618*618*618*2|^0 = 715*204008310+126 by Lm11,Lm1;
        then 2|^(11*4+0),126 are_congruent_mod 715 &
          126*126*126*126*2|^2 = 715*1410055+179 by Lem1,Lm2;
        then 2|^(44*4+2),179 are_congruent_mod 715 &
          179*179*179*179*2|^3 = 715*11486720+648 by LemS,Lm3;
        then 2|^(178*4+3),648 are_congruent_mod 715 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 716;
        2|^11 = 716*2+616 &
          616*616*616*616*2|^0 = 716*201098960+576 by Lm11,Lm1;
        then 2|^(11*4+0),576 are_congruent_mod 716 &
          576*576*576*576*2|^3 = 716*1229891778+360 by Lem1,Lm3;
        then 2|^(44*4+3),360 are_congruent_mod 716 &
          360*360*360*360*2|^0 = 716*23458324+16 by LemS,Lm1;
        then 2|^(179*4+0),16 are_congruent_mod 716 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 717;
        2|^11 = 717*2+614 &
          614*614*614*614*2|^0 = 717*198223129+523 by Lm11,Lm1;
        then 2|^(11*4+0),523 are_congruent_mod 717 &
          523*523*523*523*2|^3 = 717*834790670+338 by Lem1,Lm3;
        then 2|^(44*4+3),338 are_congruent_mod 717 &
          338*338*338*338*2|^1 = 717*36406392+8 by LemS;
        then 2|^(179*4+1),8 are_congruent_mod 717 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 718;
        2|^11 = 718*2+612 &
          612*612*612*612*2|^0 = 718*195380512+320 by Lm11,Lm1;
        then 2|^(11*4+0),320 are_congruent_mod 718 &
          320*320*320*320*2|^3 = 718*116832980+360 by Lem1,Lm3;
        then 2|^(44*4+3),360 are_congruent_mod 718 &
          360*360*360*360*2|^2 = 718*93571922+4 by LemS,Lm2;
        then 2|^(179*4+2),4 are_congruent_mod 718 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 719;
        hence thesis by XPRIMES1:719;
      end;
      suppose A: n = 720;
        2|^11 = 720*2+608 &
          608*608*608*608*2|^1 = 720*379587424+512 by Lm11;
        then 2|^(11*4+1),512 are_congruent_mod 720 &
          512*512*512*512*2|^0 = 720*95443717+496 by Lem1,Lm1;
        then 2|^(45*4+0),496 are_congruent_mod 720 &
          496*496*496*496*2|^0 = 720*84060933+496 by LemS;
        then 2|^(180*4+0),496 are_congruent_mod 720 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 721;
        2|^11 = 721*2+606 &
          606*606*606*606*2|^1 = 721*374097863+169 by Lm11;
        then 2|^(11*4+1),169 are_congruent_mod 721 &
          169*169*169*169*2|^0 = 721*1131387+694 by Lem1,Lm1;
        then 2|^(45*4+0),694 are_congruent_mod 721 &
          694*694*694*694*2|^1 = 721*643476384+128 by LemS;
        then 2|^(180*4+1),128 are_congruent_mod 721 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 722;
        2|^11 = 722*2+604 &
          604*604*604*604*2|^1 = 722*368672337+398 by Lm11;
        then 2|^(11*4+1),398 are_congruent_mod 722 &
          398*398*398*398*2|^0 = 722*34753223+210 by Lem1,Lm1;
        then 2|^(45*4+0),210 are_congruent_mod 722 &
          210*210*210*210*2|^2 = 722*10774570+460 by LemS,Lm2;
        then 2|^(180*4+2),460 are_congruent_mod 722 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 723;
        2|^11 = 723*2+602 &
          602*602*602*602*2|^1 = 723*363310260+452 by Lm11;
        then 2|^(11*4+1),452 are_congruent_mod 723 &
          452*452*452*452*2|^0 = 723*57731845+481 by Lem1,Lm1;
        then 2|^(45*4+0),481 are_congruent_mod 723 &
          481*481*481*481*2|^3 = 723*592286720+8 by LemS,Lm3;
        then 2|^(180*4+3),8 are_congruent_mod 723 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 724;
        2|^11 = 724*2+600 &
          600*600*600*600*2|^1 = 724*358011049+524 by Lm11;
        then 2|^(11*4+1),524 are_congruent_mod 724 &
          524*524*524*524*2|^1 = 724*208265137+364 by Lem1;
        then 2|^(45*4+1),364 are_congruent_mod 724 &
          364*364*364*364*2|^0 = 724*24247500+16 by LemS,Lm1;
        then 2|^(181*4+0),16 are_congruent_mod 724 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 725;
        2|^11 = 725*2+598 &
          598*598*598*598*2|^1 = 725*352774126+282 by Lm11;
        then 2|^(11*4+1),282 are_congruent_mod 725 &
          282*282*282*282*2|^1 = 725*17445700+652 by Lem1;
        then 2|^(45*4+1),652 are_congruent_mod 725 &
          652*652*652*652*2|^1 = 725*498519753+707 by LemS;
        then 2|^(181*4+1),707 are_congruent_mod 725 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 726;
        2|^11 = 726*2+596 &
          596*596*596*596*2|^1 = 726*347598916+296 by Lm11;
        then 2|^(11*4+1),296 are_congruent_mod 726 &
          296*296*296*296*2|^1 = 726*21147557+530 by Lem1;
        then 2|^(45*4+1),530 are_congruent_mod 726 &
          530*530*530*530*2|^2 = 726*434737245+130 by LemS,Lm2;
        then 2|^(181*4+2),130 are_congruent_mod 726 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 727;
        hence thesis by XPRIMES1:727;
      end;
      suppose A: n = 728;
        2|^11 = 728*2+592 &
          592*592*592*592*2|^1 = 728*337431360+512 by Lm11;
        then 2|^(11*4+1),512 are_congruent_mod 728 &
          512*512*512*512*2|^2 = 728*377579542+368 by Lem1,Lm2;
        then 2|^(45*4+2),368 are_congruent_mod 728 &
          368*368*368*368*2|^0 = 728*25191840+256 by LemS,Lm1;
        then 2|^(182*4+0),256 are_congruent_mod 728 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 729;
        2|^11 = 729*2+590 &
          590*590*590*590*2|^1 = 729*332437887+377 by Lm11;
        then 2|^(11*4+1),377 are_congruent_mod 729 &
          377*377*377*377*2|^2 = 729*110840343+517 by Lem1,Lm2;
        then 2|^(45*4+2),517 are_congruent_mod 729 &
          517*517*517*517*2|^1 = 729*196003866+728 by LemS;
        then 2|^(182*4+1),728 are_congruent_mod 729 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 730;
        2|^11 = 730*2+588 &
          588*588*588*588*2|^1 = 730*327503872+512 by Lm11;
        then 2|^(11*4+1),512 are_congruent_mod 730 &
          512*512*512*512*2|^2 = 730*376545078+4 by Lem1,Lm2;
        then 2|^(45*4+2),4 are_congruent_mod 730 &
          4*4*4*4*2|^2 = 730*1+294 by LemS;
        then 2|^(182*4+2),294 are_congruent_mod 730 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 731;
        2|^11 = 731*2+586 &
          586*586*586*586*2|^1 = 731*322628762+610 by Lm11;
        then 2|^(11*4+1),610 are_congruent_mod 731 &
          610*610*610*610*2|^2 = 731*757638358+302 by Lem1,Lm2;
        then 2|^(45*4+2),302 are_congruent_mod 731 &
          302*302*302*302*2|^3 = 731*91033320+8 by LemS,Lm3;
        then 2|^(182*4+3),8 are_congruent_mod 731 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 732;
        2|^11 = 732*2+584 &
          584*584*584*584*2|^1 = 732*317812008+416 by Lm11;
        then 2|^(11*4+1),416 are_congruent_mod 732 &
          416*416*416*416*2|^3 = 732*327304690+8 by Lem1,Lm3;
        then 2|^(45*4+3),8 are_congruent_mod 732 &
          8*8*8*8*2|^0 = 732*5+436 by LemS,Lm1;
        then 2|^(183*4+0),436 are_congruent_mod 732 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 733;
        hence thesis by XPRIMES1:733;
      end;
      suppose A: n = 734;
        2|^11 = 734*2+580 &
          580*580*580*580*2|^1 = 734*308351389+474 by Lm11;
        then 2|^(11*4+1),474 are_congruent_mod 734 &
          474*474*474*474*2|^3 = 734*550183160+368 by Lem1,Lm3;
        then 2|^(45*4+3),368 are_congruent_mod 734 &
          368*368*368*368*2|^2 = 734*99943650+4 by LemS,Lm2;
        then 2|^(183*4+2),4 are_congruent_mod 734 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 735;
        2|^11 = 735*2+578 &
          578*578*578*578*2|^1 = 735*303706446+302 by Lm11;
        then 2|^(11*4+1),302 are_congruent_mod 735 &
          302*302*302*302*2|^3 = 735*90537900+428 by Lem1,Lm3;
        then 2|^(45*4+3),428 are_congruent_mod 735 &
          428*428*428*428*2|^3 = 735*365239486+638 by LemS;
        then 2|^(183*4+3),638 are_congruent_mod 735 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 736;
        2|^11 = 736*2+576 &
          576*576*576*576*2|^2 = 736*598235403+96 by Lm11,Lm2;
        then 2|^(11*4+2),96 are_congruent_mod 736 &
          96*96*96*96*2|^0 = 736*115400+256 by Lem1,Lm1;
        then 2|^(46*4+0),256 are_congruent_mod 736 &
          256*256*256*256*2|^0 = 736*5835553+288 by LemS;
        then 2|^(184*4+0),288 are_congruent_mod 736 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 737;
        2|^11 = 737*2+574 &
          574*574*574*574*2|^2 = 737*589169251+317 by Lm11,Lm2;
        then 2|^(11*4+2),317 are_congruent_mod 737 &
          317*317*317*317*2|^0 = 737*13701545+456 by Lem1,Lm1;
        then 2|^(46*4+0),456 are_congruent_mod 737 &
          456*456*456*456*2|^1 = 737*117333460+172 by LemS;
        then 2|^(184*4+1),172 are_congruent_mod 737 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 738;
        2|^11 = 738*2+572 &
          572*572*572*572*2|^2 = 738*580213386+556 by Lm11,Lm2;
        then 2|^(11*4+2),556 are_congruent_mod 738 &
          556*556*556*556*2|^0 = 738*129491960+16 by Lem1,Lm1;
        then 2|^(46*4+0),16 are_congruent_mod 738 &
          16*16*16*16*2|^2 = 738*355+154 by LemS,Lm2;
        then 2|^(184*4+2),154 are_congruent_mod 738 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 739;
        hence thesis by XPRIMES1:739;
      end;
      suppose A: n = 740;
        2|^11 = 740*2+568 &
          568*568*568*568*2|^2 = 740*562628353+284 by Lm11,Lm2;
        then 2|^(11*4+2),284 are_congruent_mod 740 &
          284*284*284*284*2|^1 = 740*17582136+32 by Lem1;
        then 2|^(46*4+1),32 are_congruent_mod 740 &
          32*32*32*32*2|^0 = 740*1416+736 by LemS,Lm1;
        then 2|^(185*4+0),736 are_congruent_mod 740 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 741;
        2|^11 = 741*2+566 &
          566*566*566*566*2|^2 = 741*553997121+283 by Lm11,Lm2;
        then 2|^(11*4+2),283 are_congruent_mod 741 &
          283*283*283*283*2|^1 = 741*17312410+32 by Lem1;
        then 2|^(46*4+1),32 are_congruent_mod 741 &
          32*32*32*32*2|^1 = 741*2830+122 by LemS;
        then 2|^(185*4+1),122 are_congruent_mod 741 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 742;
        2|^11 = 742*2+564 &
          564*564*564*564*2|^2 = 742*545472049+506 by Lm11,Lm2;
        then 2|^(11*4+2),506 are_congruent_mod 742 &
          506*506*506*506*2|^1 = 742*176696585+522 by Lem1;
        then 2|^(46*4+1),522 are_congruent_mod 742 &
          522*522*522*522*2|^2 = 742*400256227+590 by LemS,Lm2;
        then 2|^(185*4+2),590 are_congruent_mod 742 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 743;
        hence thesis by XPRIMES1:743;
      end;
      suppose A: n = 744;
        2|^11 = 744*2+560 &
          560*560*560*560*2|^2 = 744*528736344+64 by Lm11,Lm2;
        then 2|^(11*4+2),64 are_congruent_mod 744 &
          64*64*64*64*2|^2 = 744*90200+64 by Lem1;
        then 2|^(46*4+2),64 are_congruent_mod 744 &
          64*64*64*64*2|^0 = 744*22550+16 by LemS,Lm1;
        then 2|^(186*4+0),16 are_congruent_mod 744 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 745;
        2|^11 = 745*2+558 &
          558*558*558*558*2|^2 = 745*520523707+269 by Lm11,Lm2;
        then 2|^(11*4+2),269 are_congruent_mod 745 &
          269*269*269*269*2|^2 = 745*28113365+359 by Lem1;
        then 2|^(46*4+2),359 are_congruent_mod 745 &
          359*359*359*359*2|^1 = 745*44591442+32 by LemS;
        then 2|^(186*4+1),32 are_congruent_mod 745 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 746;
        2|^11 = 746*2+556 &
          556*556*556*556*2|^2 = 746*512413225+134 by Lm11,Lm2;
        then 2|^(11*4+2),134 are_congruent_mod 746 &
          134*134*134*134*2|^2 = 746*1728782+372 by Lem1;
        then 2|^(46*4+2),372 are_congruent_mod 746 &
          372*372*372*372*2|^2 = 746*102681670+4 by LemS;
        then 2|^(186*4+2),4 are_congruent_mod 746 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 747;
        2|^11 = 747*2+554 &
          554*554*554*554*2|^2 = 747*504403914+466 by Lm11,Lm2;
        then 2|^(11*4+2),466 are_congruent_mod 747 &
          466*466*466*466*2|^2 = 747*252512601+397 by Lem1;
        then 2|^(46*4+2),397 are_congruent_mod 747 &
          397*397*397*397*2|^3 = 747*266030488+512 by LemS,Lm3;
        then 2|^(186*4+3),512 are_congruent_mod 747 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 748;
        2|^11 = 748*2+552 &
          552*552*552*552*2|^2 = 748*496494800+64 by Lm11,Lm2;
        then 2|^(11*4+2),64 are_congruent_mod 748 &
          64*64*64*64*2|^3 = 748*179435+348 by Lem1,Lm3;
        then 2|^(46*4+3),348 are_congruent_mod 748 &
          348*348*348*348*2|^0 = 748*19607190+696 by LemS,Lm1;
        then 2|^(187*4+0),696 are_congruent_mod 748 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 749;
        2|^11 = 749*2+550 &
          550*550*550*550*2|^2 = 749*488684913+163 by Lm11,Lm2;
        then 2|^(11*4+2),163 are_congruent_mod 749 &
          163*163*163*163*2|^3 = 749*7539778+366 by Lem1,Lm3;
        then 2|^(46*4+3),366 are_congruent_mod 749 &
          366*366*366*366*2|^1 = 749*47915113+235 by LemS;
        then 2|^(187*4+1),235 are_congruent_mod 749 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 750;
        2|^11 = 750*2+548 &
          548*548*548*548*2|^2 = 750*480973292+664 by Lm11,Lm2;
        then 2|^(11*4+2),664 are_congruent_mod 750 &
          664*664*664*664*2|^3 = 750*2073485683+278 by Lem1,Lm3;
        then 2|^(46*4+3),278 are_congruent_mod 750 &
          278*278*278*278*2|^2 = 750*31855022+124 by LemS,Lm2;
        then 2|^(187*4+2),124 are_congruent_mod 750 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 751;
        hence thesis by XPRIMES1:751;
      end;
      suppose A: n = 752;
        2|^11 = 752*2+544 &
          544*544*544*544*2|^3 = 752*931682086+96 by Lm11,Lm3;
        then 2|^(11*4+3),96 are_congruent_mod 752 &
          96*96*96*96*2|^0 = 752*112945+16 by Lem1,Lm1;
        then 2|^(47*4+0),16 are_congruent_mod 752 &
          16*16*16*16*2|^0 = 752*87+112 by LemS;
        then 2|^(188*4+0),112 are_congruent_mod 752 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 753;
        2|^11 = 753*2+542 &
          542*542*542*542*2|^3 = 753*916837053+659 by Lm11,Lm3;
        then 2|^(11*4+3),659 are_congruent_mod 753 &
          659*659*659*659*2|^0 = 753*250464790+91 by Lem1,Lm1;
        then 2|^(47*4+0),91 are_congruent_mod 753 &
          91*91*91*91*2|^1 = 753*182138+8 by LemS;
        then 2|^(188*4+1),8 are_congruent_mod 753 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 754;
        2|^11 = 754*2+540 &
          540*540*540*540*2|^3 = 754*902181007+722 by Lm11,Lm3;
        then 2|^(11*4+3),722 are_congruent_mod 754 &
          722*722*722*722*2|^0 = 754*360393910+516 by Lem1,Lm1;
        then 2|^(47*4+0),516 are_congruent_mod 754 &
          516*516*516*516*2|^2 = 754*376086246+660 by LemS,Lm2;
        then 2|^(188*4+2),660 are_congruent_mod 754 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 755;
        2|^11 = 755*2+538 &
          538*538*538*538*2|^3 = 755*887712096+608 by Lm11,Lm3;
        then 2|^(11*4+3),608 are_congruent_mod 755 &
          608*608*608*608*2|^0 = 755*180995328+256 by Lem1,Lm1;
        then 2|^(47*4+0),256 are_congruent_mod 755 &
          256*256*256*256*2|^3 = 755*45509587+183 by LemS,Lm3;
        then 2|^(188*4+3),183 are_congruent_mod 755 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 756;
        2|^11 = 756*2+536 &
          536*536*536*536*2|^3 = 756*873428482+536 by Lm11,Lm3;
        then 2|^(11*4+3),536 are_congruent_mod 756 &
          536*536*536*536*2|^1 = 756*218357120+512 by Lem1;
        then 2|^(47*4+1),512 are_congruent_mod 756 &
          512*512*512*512*2|^0 = 756*90898778+568 by LemS,Lm1;
        then 2|^(189*4+0),568 are_congruent_mod 756 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 757;
        hence thesis by XPRIMES1:757;
      end;
      suppose A: n = 758;
        2|^11 = 758*2+532 &
          532*532*532*532*2|^3 = 758*845409863+454 by Lm11,Lm3;
        then 2|^(11*4+3),454 are_congruent_mod 758 &
          454*454*454*454*2|^1 = 758*112094473+378 by Lem1;
        then 2|^(47*4+1),378 are_congruent_mod 758 &
          378*378*378*378*2|^2 = 758*107735290+4 by LemS,Lm2;
        then 2|^(189*4+2),4 are_congruent_mod 758 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 759;
        2|^11 = 759*2+530 &
          530*530*530*530*2|^3 = 759*831671251+491 by Lm11,Lm3;
        then 2|^(11*4+3),491 are_congruent_mod 759 &
          491*491*491*491*2|^1 = 759*153149008+50 by Lem1;
        then 2|^(47*4+1),50 are_congruent_mod 759 &
          50*50*50*50*2|^3 = 759*65876+116 by LemS,Lm3;
        then 2|^(189*4+3),116 are_congruent_mod 759 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 760;
        2|^11 = 760*2+528 &
          528*528*528*528*2|^3 = 760*818110722+528 by Lm11,Lm3;
        then 2|^(11*4+3),528 are_congruent_mod 760 &
          528*528*528*528*2|^2 = 760*409055361+264 by Lem1,Lm2;
        then 2|^(47*4+2),264 are_congruent_mod 760 &
          264*264*264*264*2|^0 = 760*6391490+16 by LemS,Lm1;
        then 2|^(190*4+0),16 are_congruent_mod 760 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 761;
        hence thesis by XPRIMES1:761;
      end;
      suppose A: n = 762;
        2|^11 = 762*2+524 &
          524*524*524*524*2|^3 = 762*791516848+32 by Lm11,Lm3;
        then 2|^(11*4+3),32 are_congruent_mod 762 &
          32*32*32*32*2|^2 = 762*5504+256 by Lem1,Lm2;
        then 2|^(47*4+2),256 are_congruent_mod 762 &
          256*256*256*256*2|^2 = 762*22545760+64 by LemS;
        then 2|^(190*4+2),64 are_congruent_mod 762 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 763;
        2|^11 = 763*2+522 &
          522*522*522*522*2|^3 = 763*778480002+522 by Lm11,Lm3;
        then 2|^(11*4+3),522 are_congruent_mod 763 &
          522*522*522*522*2|^2 = 763*389240001+261 by Lem1,Lm2;
        then 2|^(47*4+2),261 are_congruent_mod 763 &
          261*261*261*261*2|^3 = 763*48655000+128 by LemS,Lm3;
        then 2|^(190*4+3),128 are_congruent_mod 763 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 764;
        2|^11 = 764*2+520 &
          520*520*520*520*2|^3 = 764*765614240+640 by Lm11,Lm3;
        then 2|^(11*4+3),640 are_congruent_mod 764 &
          640*640*640*640*2|^3 = 764*1756776544+384 by Lem1;
        then 2|^(47*4+3),384 are_congruent_mod 764 &
          384*384*384*384*2|^0 = 764*28459780+16 by LemS,Lm1;
        then 2|^(191*4+0),16 are_congruent_mod 764 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 765;
        2|^11 = 765*2+518 &
          518*518*518*518*2|^3 = 765*752917845+383 by Lm11,Lm3;
        then 2|^(11*4+3),383 are_congruent_mod 765 &
          383*383*383*383*2|^3 = 765*225021309+383 by Lem1;
        then 2|^(47*4+3),383 are_congruent_mod 765 &
          383*383*383*383*2|^1 = 765*56255327+287 by LemS;
        then 2|^(191*4+1),287 are_congruent_mod 765 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 766;
        2|^11 = 766*2+516 &
          516*516*516*516*2|^3 = 766*740389112+496 by Lm11,Lm3;
        then 2|^(11*4+3),496 are_congruent_mod 766 &
          496*496*496*496*2|^3 = 766*632103104+384 by Lem1;
        then 2|^(47*4+3),384 are_congruent_mod 766 &
          384*384*384*384*2|^2 = 766*113541890+4 by LemS,Lm2;
        then 2|^(191*4+2),4 are_congruent_mod 766 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 767;
        2|^11 = 767*2+514 &
          514*514*514*514*2|^3 = 767*728026351+111 by Lm11,Lm3;
        then 2|^(11*4+3),111 are_congruent_mod 767 &
          111*111*111*111*2|^3 = 767*1583385+33 by Lem1;
        then 2|^(47*4+3),33 are_congruent_mod 767 &
          33*33*33*33*2|^3 = 767*12369+345 by LemS;
        then 2|^(191*4+3),345 are_congruent_mod 767 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 768;
        2|^12 = 768*5+256 &
          256*256*256*256*2|^0 = 768*5592405+256 by Lm12,Lm1;
        then 2|^(12*4+0),256 are_congruent_mod 768 &
          256*256*256*256*2|^0 = 768*5592405+256 by Lem1;
        then 2|^(48*4+0),256 are_congruent_mod 768 &
          256*256*256*256*2|^0 = 768*5592405+256 by LemS;
        then 2|^(192*4+0),256 are_congruent_mod 768 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 769;
        hence thesis by XPRIMES1:769;
      end;
      suppose A: n = 770;
        2|^12 = 770*5+246 & 246*246*246*246*2|^0 = 770*4756086+36 by Lm12,Lm1;
        then 2|^(12*4+0),36 are_congruent_mod 770 &
          36*36*36*36*2|^0 = 770*2181+246 by Lem1;
        then 2|^(48*4+0),246 are_congruent_mod 770 &
          246*246*246*246*2|^2 = 770*19024344+144 by LemS,Lm2;
        then 2|^(192*4+2),144 are_congruent_mod 770 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 771;
        2|^12 = 771*5+241 & 241*241*241*241*2|^0 = 771*4375360+1 by Lm12,Lm1;
        then 2|^(12*4+0),1 are_congruent_mod 771 &
          1*1*1*1*2|^0 = 771*0+1 by Lem1;
        then 2|^(48*4+0),1 are_congruent_mod 771 &
          1*1*1*1*2|^3 = 771*0+8 by LemS,Lm3;
        then 2|^(192*4+3),8 are_congruent_mod 771 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 772;
        2|^12 = 772*5+236 &
          236*236*236*236*2|^0 = 772*4018192+192 by Lm12,Lm1;
        then 2|^(12*4+0),192 are_congruent_mod 772 &
          192*192*192*192*2|^1 = 772*3520607+388 by Lem1;
        then 2|^(48*4+1),388 are_congruent_mod 772 &
          388*388*388*388*2|^0 = 772*29356860+16 by LemS,Lm1;
        then 2|^(193*4+0),16 are_congruent_mod 772 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 773;
        hence thesis by XPRIMES1:773;
      end;
      suppose A: n = 774;
        2|^12 = 774*5+226 & 226*226*226*226*2|^0 = 774*3370488+64 by Lm12,Lm1;
        then 2|^(12*4+0),64 are_congruent_mod 774 &
          64*64*64*64*2|^1 = 774*43351+758 by Lem1;
        then 2|^(48*4+1),758 are_congruent_mod 774 &
          758*758*758*758*2|^2 = 774*1706066098+532 by LemS,Lm2;
        then 2|^(193*4+2),532 are_congruent_mod 774 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 775;
        2|^12 = 775*5+221 &
          221*221*221*221*2|^0 = 775*3077991+256 by Lm12,Lm1;
        then 2|^(12*4+0),256 are_congruent_mod 775 &
          256*256*256*256*2|^1 = 775*11083786+442 by Lem1;
        then 2|^(48*4+1),442 are_congruent_mod 775 &
          442*442*442*442*2|^3 = 775*393982890+218 by LemS,Lm3;
        then 2|^(193*4+3),218 are_congruent_mod 775 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 776;
        2|^12 = 776*5+216 &
          216*216*216*216*2|^0 = 776*2805131+680 by Lm12,Lm1;
        then 2|^(12*4+0),680 are_congruent_mod 776 &
          680*680*680*680*2|^2 = 776*1102132783+392 by Lem1,Lm2;
        then 2|^(48*4+2),392 are_congruent_mod 776 &
          392*392*392*392*2|^0 = 776*30428640+256 by LemS,Lm1;
        then 2|^(194*4+0),256 are_congruent_mod 776 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 777;
        2|^12 = 777*5+211 &
          211*211*211*211*2|^0 = 777*2550990+211 by Lm12,Lm1;
        then 2|^(12*4+0),211 are_congruent_mod 777 &
          211*211*211*211*2|^2 = 777*10203961+67 by Lem1,Lm2;
        then 2|^(48*4+2),67 are_congruent_mod 777 &
          67*67*67*67*2|^1 = 777*51869+29 by LemS;
        then 2|^(194*4+1),29 are_congruent_mod 777 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 778;
        2|^12 = 778*5+206 & 206*206*206*206*2|^0 = 778*2314671+58 by Lm12,Lm1;
        then 2|^(12*4+0),58 are_congruent_mod 778 &
          58*58*58*58*2|^2 = 778*58182+388 by Lem1,Lm2;
        then 2|^(48*4+2),388 are_congruent_mod 778 &
          388*388*388*388*2|^2 = 778*116521830+4 by LemS;
        then 2|^(194*4+2),4 are_congruent_mod 778 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 779;
        2|^12 = 779*5+201 &
          201*201*201*201*2|^0 = 779*2095302+543 by Lm12,Lm1;
        then 2|^(12*4+0),543 are_congruent_mod 779 &
          543*543*543*543*2|^2 = 779*446397601+25 by Lem1,Lm2;
        then 2|^(48*4+2),25 are_congruent_mod 779 &
          25*25*25*25*2|^3 = 779*4011+431 by LemS,Lm3;
        then 2|^(194*4+3),431 are_congruent_mod 779 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 780;
        2|^12 = 780*5+196 &
          196*196*196*196*2|^0 = 780*1892037+196 by Lm12,Lm1;
        then 2|^(12*4+0),196 are_congruent_mod 780 &
          196*196*196*196*2|^3 = 780*15136298+8 by Lem1,Lm3;
        then 2|^(48*4+3),8 are_congruent_mod 780 &
          8*8*8*8*2|^0 = 780*5+196 by LemS,Lm1;
        then 2|^(195*4+0),196 are_congruent_mod 780 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 781;
        2|^12 = 781*5+191 &
          191*191*191*191*2|^0 = 781*1704050+311 by Lm12,Lm1;
        then 2|^(12*4+0),311 are_congruent_mod 781 &
          311*311*311*311*2|^3 = 781*95825370+758 by Lem1,Lm3;
        then 2|^(48*4+3),758 are_congruent_mod 781 &
          758*758*758*758*2|^1 = 781*845387426+486 by LemS;
        then 2|^(195*4+1),486 are_congruent_mod 781 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 782;
        2|^12 = 782*5+186 &
          186*186*186*186*2|^0 = 782*1530541+154 by Lm12,Lm1;
        then 2|^(12*4+0),154 are_congruent_mod 782 &
          154*154*154*154*2|^3 = 782*5753950+348 by Lem1,Lm3;
        then 2|^(48*4+3),348 are_congruent_mod 782 &
          348*348*348*348*2|^2 = 782*75018817+370 by LemS,Lm2;
        then 2|^(195*4+2),370 are_congruent_mod 782 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 783;
        2|^12 = 783*5+181 &
          181*181*181*181*2|^0 = 783*1370731+748 by Lm12,Lm1;
        then 2|^(12*4+0),748 are_congruent_mod 783 &
          748*748*748*748*2|^3 = 783*(42892*74569)+44 by Lem1,Lm3;
        then 2|^(48*4+3),44 are_congruent_mod 783 &
          44*44*44*44*2|^3 = 783*38294+566 by LemS;
        then 2|^(195*4+3),566 are_congruent_mod 783 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 784;
        2|^12 = 784*5+176 &
          176*176*176*176*2|^1 = 784*2447736+128 by Lm12;
        then 2|^(12*4+1),128 are_congruent_mod 784 &
          128*128*128*128*2|^0 = 784*342392+128 by Lem1,Lm1;
        then 2|^(49*4+0),128 are_congruent_mod 784 &
          128*128*128*128*2|^0 = 784*342392+128 by LemS;
        then 2|^(196*4+0),128 are_congruent_mod 784 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 785;
        2|^12 = 785*5+171 &
          171*171*171*171*2|^1 = 785*2178435+687 by Lm12;
        then 2|^(12*4+1),687 are_congruent_mod 785 &
          687*687*687*687*2|^0 = 785*283763996+101 by Lem1,Lm1;
        then 2|^(49*4+0),101 are_congruent_mod 785 &
          101*101*101*101*2|^1 = 785*265122+32 by LemS;
        then 2|^(196*4+1),32 are_congruent_mod 785 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 786;
        2|^12 = 786*5+166 &
          166*166*166*166*2|^1 = 786*1932145+302 by Lm12;
        then 2|^(12*4+1),302 are_congruent_mod 786 &
          302*302*302*302*2|^0 = 786*10582912+784 by Lem1,Lm1;
        then 2|^(49*4+0),784 are_congruent_mod 786 &
          784*784*784*784*2|^2 = 786*1922656480+64 by LemS,Lm2;
        then 2|^(196*4+2),64 are_congruent_mod 786 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 787;
        hence thesis by XPRIMES1:787;
      end;
      suppose A: n = 788;
        2|^12 = 788*5+156 &
          156*156*156*156*2|^1 = 788*1503149+380 by Lm12;
        then 2|^(12*4+1),380 are_congruent_mod 788 &
          380*380*380*380*2|^1 = 788*52922233+396 by Lem1;
        then 2|^(49*4+1),396 are_congruent_mod 788 &
          396*396*396*396*2|^0 = 788*31207180+16 by LemS,Lm1;
        then 2|^(197*4+0),16 are_congruent_mod 788 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 789;
        2|^12 = 789*5+151 &
          151*151*151*151*2|^1 = 789*1317834+176 by Lm12;
        then 2|^(12*4+1),176 are_congruent_mod 789 &
          176*176*176*176*2|^1 = 789*2432224+416 by Lem1;
        then 2|^(49*4+1),416 are_congruent_mod 789 &
          416*416*416*416*2|^1 = 789*75914776+8 by LemS;
        then 2|^(197*4+1),8 are_congruent_mod 789 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 790;
        2|^12 = 790*5+146 &
          146*146*146*146*2|^1 = 790*1150308+392 by Lm12;
        then 2|^(12*4+1),392 are_congruent_mod 790 &
          392*392*392*392*2|^1 = 790*59778797+162 by Lem1;
        then 2|^(49*4+1),162 are_congruent_mod 790 &
          162*162*162*162*2|^2 = 790*3487329+234 by LemS,Lm2;
        then 2|^(197*4+2),234 are_congruent_mod 790 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 791;
        2|^12 = 791*5+141 & 141*141*141*141*2|^1 = 791*999378+324 by Lm12;
        then 2|^(12*4+1),324 are_congruent_mod 791 &
          324*324*324*324*2|^1 = 791*27863364+228 by Lem1;
        then 2|^(49*4+1),228 are_congruent_mod 791 &
          228*228*228*228*2|^3 = 791*27330834+354 by LemS,Lm3;
        then 2|^(197*4+3),354 are_congruent_mod 791 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 792;
        2|^12 = 792*5+136 & 136*136*136*136*2|^1 = 792*863893+776 by Lm12;
        then 2|^(12*4+1),776 are_congruent_mod 792 &
          776*776*776*776*2|^2 = 792*1831393610+784 by Lem1,Lm2;
        then 2|^(49*4+2),784 are_congruent_mod 792 &
          784*784*784*784*2|^0 = 792*477022725+136 by LemS,Lm1;
        then 2|^(198*4+0),136 are_congruent_mod 792 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 793;
        2|^12 = 793*5+131 & 131*131*131*131*2|^1 = 793*742748+678 by Lm12;
        then 2|^(12*4+1),678 are_congruent_mod 793 &
          678*678*678*678*2|^2 = 793*1065873290+454 by Lem1,Lm2;
        then 2|^(49*4+2),454 are_congruent_mod 793 &
          454*454*454*454*2|^1 = 793*107147050+262 by LemS;
        then 2|^(198*4+1),262 are_congruent_mod 793 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 794;
        2|^12 = 794*5+126 & 126*126*126*126*2|^1 = 794*634880+32 by Lm12;
        then 2|^(12*4+1),32 are_congruent_mod 794 &
          32*32*32*32*2|^2 = 794*5282+396 by Lem1,Lm2;
        then 2|^(49*4+2),396 are_congruent_mod 794 &
          396*396*396*396*2|^2 = 794*123885430+4 by LemS;
        then 2|^(198*4+2),4 are_congruent_mod 794 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 795;
        2|^12 = 795*5+121 & 121*121*121*121*2|^1 = 795*539267+497 by Lm12;
        then 2|^(12*4+1),497 are_congruent_mod 795 &
          497*497*497*497*2|^2 = 795*306985892+184 by Lem1,Lm2;
        then 2|^(49*4+2),184 are_congruent_mod 795 &
          184*184*184*184*2|^3 = 795*11534377+173 by LemS,Lm3;
        then 2|^(198*4+3),173 are_congruent_mod 795 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 796;
        2|^12 = 796*5+116 & 116*116*116*116*2|^1 = 796*454934+408 by Lm12;
        then 2|^(12*4+1),408 are_congruent_mod 796 &
          408*408*408*408*2|^3 = 796*278495108+400 by Lem1,Lm3;
        then 2|^(49*4+3),400 are_congruent_mod 796 &
          400*400*400*400*2|^0 = 796*32160804+16 by LemS,Lm1;
        then 2|^(199*4+0),16 are_congruent_mod 796 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 797;
        hence thesis by XPRIMES1:797;
      end;
      suppose A: n = 798;
        2|^12 = 798*5+106 & 106*106*106*106*2|^1 = 798*316410+212 by Lm12;
        then 2|^(12*4+1),212 are_congruent_mod 798 &
          212*212*212*212*2|^3 = 798*20250257+2 by Lem1,Lm3;
        then 2|^(49*4+3),2 are_congruent_mod 798 &
          2*2*2*2*2|^2 = 798*0+64 by LemS,Lm2;
        then 2|^(199*4+2),64 are_congruent_mod 798 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 799;
        2|^12 = 799*5+101 & 101*101*101*101*2|^1 = 799*260476+478 by Lm12;
        then 2|^(12*4+1),478 are_congruent_mod 799 &
          478*478*478*478*2|^3 = 799*522702761+9 by Lem1,Lm3;
        then 2|^(49*4+3),9 are_congruent_mod 799 &
          9*9*9*9*2|^3 = 799*65+553 by LemS;
        then 2|^(199*4+3),553 are_congruent_mod 799 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 800;
        2|^12 = 800*5+96 & 96*96*96*96*2|^2 = 800*424673+224 by Lm12,Lm2;
        then 2|^(12*4+2),224 are_congruent_mod 800 &
          224*224*224*224*2|^0 = 800*3147038+576 by Lem1,Lm1;
        then 2|^(50*4+0),576 are_congruent_mod 800 &
          576*576*576*576*2|^0 = 800*137594142+576 by LemS;
        then 2|^(200*4+0),576 are_congruent_mod 800 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 801;
        2|^12 = 801*5+91 & 91*91*91*91*2|^2 = 801*342446+598 by Lm12,Lm2;
        then 2|^(12*4+2),598 are_congruent_mod 801 &
          598*598*598*598*2|^0 = 801*159651212+4 by Lem1,Lm1;
        then 2|^(50*4+0),4 are_congruent_mod 801 &
          4*4*4*4*2|^1 = 801*0+512 by LemS;
        then 2|^(200*4+1),512 are_congruent_mod 801 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 802;
        2|^12 = 802*5+86 & 86*86*86*86*2|^2 = 802*272822+20 by Lm12,Lm2;
        then 2|^(12*4+2),20 are_congruent_mod 802 &
          20*20*20*20*2|^0 = 802*199+402 by Lem1,Lm1;
        then 2|^(50*4+0),402 are_congruent_mod 802 &
          402*402*402*402*2|^2 = 802*130253630+4 by LemS,Lm2;
        then 2|^(200*4+2),4 are_congruent_mod 802 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 803;
        2|^12 = 803*5+81 & 81*81*81*81*2|^2 = 803*214429+397 by Lm12,Lm2;
        then 2|^(12*4+2),397 are_congruent_mod 803 &
          397*397*397*397*2|^0 = 803*30934740+661 by Lem1,Lm1;
        then 2|^(50*4+0),661 are_congruent_mod 803 &
          661*661*661*661*2|^3 = 803*1901867598+734 by LemS,Lm3;
        then 2|^(200*4+3),734 are_congruent_mod 803 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 804;
        2|^12 = 804*5+76 & 76*76*76*76*2|^2 = 804*165980+784 by Lm12,Lm2;
        then 2|^(12*4+2),784 are_congruent_mod 804 &
          784*784*784*784*2|^1 = 804*939805966+8 by Lem1;
        then 2|^(50*4+1),8 are_congruent_mod 804 &
          8*8*8*8*2|^0 = 804*5+76 by LemS,Lm1;
        then 2|^(201*4+0),76 are_congruent_mod 804 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 804 < n & n <= 904;
      then 804+1 <= n & n <= 904 by NAT_1:13;
      then n = 805 or ... or n = 904;
      then per cases;
      suppose A: n = 805;
        2|^12 = 805*5+71 & 71*71*71*71*2|^2 = 805*126269+179 by Lm12,Lm2;
        then 2|^(12*4+2),179 are_congruent_mod 805 &
          179*179*179*179*2|^1 = 805*2550622+652 by Lem1;
        then 2|^(50*4+1),652 are_congruent_mod 805 &
          652*652*652*652*2|^1 = 805*448977418+142 by LemS;
        then 2|^(201*4+1),142 are_congruent_mod 805 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 806;
        2|^12 = 806*5+66 & 66*66*66*66*2|^2 = 806*94167+342 by Lm12,Lm2;
        then 2|^(12*4+2),342 are_congruent_mod 806 &
          342*342*342*342*2|^1 = 806*33946841+746 by Lem1;
        then 2|^(50*4+1),746 are_congruent_mod 806 &
          746*746*746*746*2|^2 = 806*1537022621+498 by LemS,Lm2;
        then 2|^(201*4+2),498 are_congruent_mod 806 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 807;
        2|^12 = 807*5+61 & 61*61*61*61*2|^2 = 807*68628+568 by Lm12,Lm2;
        then 2|^(12*4+2),568 are_congruent_mod 807 &
          568*568*568*568*2|^1 = 807*257958476+620 by Lem1;
        then 2|^(50*4+1),620 are_congruent_mod 807 &
          620*620*620*620*2|^3 = 807*1464816456+8 by LemS,Lm3;
        then 2|^(201*4+3),8 are_congruent_mod 807 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 808;
        2|^12 = 808*5+56 & 56*56*56*56*2|^2 = 808*48685+504 by Lm12,Lm2;
        then 2|^(12*4+2),504 are_congruent_mod 808 &
          504*504*504*504*2|^2 = 808*319426377+408 by Lem1;
        then 2|^(50*4+2),408 are_congruent_mod 808 &
          408*408*408*408*2|^0 = 808*34294880+256 by LemS,Lm1;
        then 2|^(202*4+0),256 are_congruent_mod 808 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 809;
        hence thesis by XPRIMES1:809;
      end;
      suppose A: n = 810;
        2|^12 = 810*5+46 & 46*46*46*46*2|^2 = 810*22110+724 by Lm12,Lm2;
        then 2|^(12*4+2),724 are_congruent_mod 810 &
          724*724*724*724*2|^2 = 810*1356841871+394 by Lem1;
        then 2|^(50*4+2),394 are_congruent_mod 810 &
          394*394*394*394*2|^2 = 810*119003534+244 by LemS;
        then 2|^(202*4+2),244 are_congruent_mod 810 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 811;
        hence thesis by XPRIMES1:811;
      end;
      suppose A: n = 812;
        2|^12 = 812*5+36 & 36*36*36*36*2|^2 = 812*8273+788 by Lm12,Lm2;
        then 2|^(12*4+2),788 are_congruent_mod 812 &
          788*788*788*788*2|^3 = 812*(3988*952541)+592 by Lem1,Lm3;
        then 2|^(50*4+3),592 are_congruent_mod 812 &
          592*592*592*592*2|^0 = 812*151262334+88 by LemS,Lm1;
        then 2|^(203*4+0),88 are_congruent_mod 812 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 813;
        2|^12 = 813*5+31 & 31*31*31*31*2|^2 = 813*4543+625 by Lm12,Lm2;
        then 2|^(12*4+2),625 are_congruent_mod 813 &
          625*625*625*625*2|^3 = 813*1501479858+446 by Lem1,Lm3;
        then 2|^(50*4+3),446 are_congruent_mod 813 &
          446*446*446*446*2|^1 = 813*97337208+8 by LemS;
        then 2|^(203*4+1),8 are_congruent_mod 813 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 814;
        2|^12 = 814*5+26 & 26*26*26*26*2|^2 = 814*2245+474 by Lm12,Lm2;
        then 2|^(12*4+2),474 are_congruent_mod 814 &
          474*474*474*474*2|^3 = 814*496111105+338 by Lem1,Lm3;
        then 2|^(50*4+3),338 are_congruent_mod 814 &
          338*338*338*338*2|^2 = 814*64136076+280 by LemS,Lm2;
        then 2|^(203*4+2),280 are_congruent_mod 814 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 815;
        2|^12 = 815*5+21 & 21*21*21*21*2|^2 = 815*954+414 by Lm12,Lm2;
        then 2|^(12*4+2),414 are_congruent_mod 815 &
          414*414*414*414*2|^3 = 815*288359154+18 by Lem1,Lm3;
        then 2|^(50*4+3),18 are_congruent_mod 815 &
          18*18*18*18*2|^3 = 815*1030+358 by LemS;
        then 2|^(203*4+3),358 are_congruent_mod 815 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 816;
        2|^12 = 816*5+16 & 16*16*16*16*2|^3 = 816*642+416 by Lm12,Lm3;
        then 2|^(12*4+3),416 are_congruent_mod 816 &
          416*416*416*416*2|^0 = 816*36701445+16 by Lem1,Lm1;
        then 2|^(51*4+0),16 are_congruent_mod 816 &
          16*16*16*16*2|^0 = 816*80+256 by LemS;
        then 2|^(204*4+0),256 are_congruent_mod 816 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 817;
        2|^12 = 817*5+11 & 11*11*11*11*2|^3 = 817*143+297 by Lm12,Lm3;
        then 2|^(12*4+3),297 are_congruent_mod 817 &
          297*297*297*297*2|^0 = 817*9523656+729 by Lem1,Lm1;
        then 2|^(51*4+0),729 are_congruent_mod 817 &
          729*729*729*729*2|^1 = 817*691381974+204 by LemS;
        then 2|^(204*4+1),204 are_congruent_mod 817 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 818;
        2|^12 = 818*5+6 & 6*6*6*6*2|^3 = 818*12+552 by Lm12,Lm3;
        then 2|^(12*4+3),552 are_congruent_mod 818 &
          552*552*552*552*2|^0 = 818*113501867+410 by Lem1,Lm1;
        then 2|^(51*4+0),410 are_congruent_mod 818 &
          410*410*410*410*2|^2 = 818*138179022+4 by LemS,Lm2;
        then 2|^(204*4+2),4 are_congruent_mod 818 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 819;
        2|^12 = 819*5+1 & 1*1*1*1*2|^3 = 819*0+8 by Lm12,Lm3;
        then 2|^(12*4+3),8 are_congruent_mod 819 &
          8*8*8*8*2|^0 = 819*5+1 by Lem1,Lm1;
        then 2|^(51*4+0),1 are_congruent_mod 819 &
          1*1*1*1*2|^3 = 819*0+8 by LemS,Lm3;
        then 2|^(204*4+3),8 are_congruent_mod 819 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 820;
        2|^12 = 820*4+816 &
          816*816*816*816*2|^3 = 820*(23358*185183)+408 by Lm12,Lm3;
        then 2|^(12*4+3),408 are_congruent_mod 820 &
          408*408*408*408*2|^1 = 820*67586008+32 by Lem1;
        then 2|^(51*4+1),32 are_congruent_mod 820 &
          32*32*32*32*2|^0 = 820*1278+616 by LemS,Lm1;
        then 2|^(205*4+0),616 are_congruent_mod 820 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 821;
        hence thesis by XPRIMES1:821;
      end;
      suppose A: n = 822;
        2|^12 = 822*4+808 &
          808*808*808*808*2|^3 = 822*(3177*1305709)+722 by Lm12,Lm3;
        then 2|^(12*4+3),722 are_congruent_mod 822 &
          722*722*722*722*2|^1 = 822*661160605+2 by Lem1;
        then 2|^(51*4+1),2 are_congruent_mod 822 &
          2*2*2*2*2|^2 = 822*0+64 by LemS,Lm2;
        then 2|^(205*4+2),64 are_congruent_mod 822 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 823;
        hence thesis by XPRIMES1:823;
      end;
      suppose A: n = 824;
        2|^12 = 824*4+800 &
          800*800*800*800*2|^3 = 824*(3*1325566343)+104 by Lm12,Lm3;
        then 2|^(12*4+3),104 are_congruent_mod 824 &
          104*104*104*104*2|^2 = 824*567892+416 by Lem1,Lm2;
        then 2|^(51*4+2),416 are_congruent_mod 824 &
          416*416*416*416*2|^0 = 824*36345120+256 by LemS,Lm1;
        then 2|^(206*4+0),256 are_congruent_mod 824 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 825;
        2|^12 = 825*4+796 &
          796*796*796*796*2|^3 = 825*(1990*1956299)+398 by Lm12,Lm3;
        then 2|^(12*4+3),398 are_congruent_mod 825 &
          398*398*398*398*2|^2 = 825*121657344+64 by Lem1,Lm2;
        then 2|^(51*4+2),64 are_congruent_mod 825 &
          64*64*64*64*2|^1 = 825*40672+32 by LemS;
        then 2|^(206*4+1),32 are_congruent_mod 825 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 826;
        2|^12 = 826*4+792 &
          792*792*792*792*2|^3 = 826*(36506*104387)+596 by Lm12,Lm3;
        then 2|^(12*4+3),596 are_congruent_mod 826 &
          596*596*596*596*2|^2 = 826*611033446+228 by Lem1,Lm2;
        then 2|^(51*4+2),228 are_congruent_mod 826 &
          228*228*228*228*2|^2 = 826*13086374+100 by LemS;
        then 2|^(206*4+2),100 are_congruent_mod 826 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 827;
        hence thesis by XPRIMES1:827;
      end;
      suppose A: n = 828;
        2|^12 = 828*4+784 &
          784*784*784*784*2|^3 = 828*(60417*60417+46964)+404 by Lm12,Lm3;
        then 2|^(12*4+3),404 are_congruent_mod 828 &
          404*404*404*404*2|^3 = 828*257386112+512 by Lem1;
        then 2|^(51*4+3),512 are_congruent_mod 828 &
          512*512*512*512*2|^0 = 828*82994537+100 by LemS,Lm1;
        then 2|^(207*4+0),100 are_congruent_mod 828 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 829;
        hence thesis by XPRIMES1:829;
      end;
      suppose A: n = 830;
        2|^12 = 830*4+776 &
          776*776*776*776*2|^3 = 830*(27467*127247)+138 by Lm12,Lm3;
        then 2|^(12*4+3),138 are_congruent_mod 830 &
          138*138*138*138*2|^3 = 830*3495652+328 by Lem1;
        then 2|^(51*4+3),328 are_congruent_mod 830 &
          328*328*328*328*2|^2 = 830*55779841+194 by LemS,Lm2;
        then 2|^(207*4+2),194 are_congruent_mod 830 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 831;
        2|^12 = 831*4+772 &
          772*772*772*772*2|^3 = 831*(279*12256147)+245 by Lm12,Lm3;
        then 2|^(12*4+3),245 are_congruent_mod 831 &
          245*245*245*245*2|^3 = 831*34685926+494 by Lem1;
        then 2|^(51*4+3),494 are_congruent_mod 831 &
          494*494*494*494*2|^3 = 831*573319560+8 by LemS;
        then 2|^(207*4+3),8 are_congruent_mod 831 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 832;
        2|^13 = 832*9+704 &
          704*704*704*704*2|^0 = 832*295234638+640 by Lm13,Lm1;
        then 2|^(13*4+0),640 are_congruent_mod 832 &
          640*640*640*640*2|^0 = 832*201649230+640 by Lem1;
        then 2|^(52*4+0),640 are_congruent_mod 832 &
          640*640*640*640*2|^0 = 832*201649230+640 by LemS;
        then 2|^(208*4+0),640 are_congruent_mod 832 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 833;
        2|^13 = 833*9+695 &
          695*695*695*695*2|^0 = 833*280087815+730 by Lm13,Lm1;
        then 2|^(13*4+0),730 are_congruent_mod 833 &
          730*730*730*730*2|^0 = 833*340915258+86 by Lem1;
        then 2|^(52*4+0),86 are_congruent_mod 833 &
          86*86*86*86*2|^1 = 833*131334+410 by LemS;
        then 2|^(208*4+1),410 are_congruent_mod 833 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 834;
        2|^13 = 834*9+686 &
          686*686*686*686*2|^0 = 834*265540282+28 by Lm13,Lm1;
        then 2|^(13*4+0),28 are_congruent_mod 834 &
          28*28*28*28*2|^0 = 834*736+832 by Lem1;
        then 2|^(52*4+0),832 are_congruent_mod 834 &
          832*832*832*832*2|^2 = 834*(45392*50630)+64 by LemS,Lm2;
        then 2|^(208*4+2),64 are_congruent_mod 834 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 835;
        2|^13 = 835*9+677 &
          677*677*677*677*2|^0 = 835*251575415+716 by Lm13,Lm1;
        then 2|^(13*4+0),716 are_congruent_mod 835 &
          716*716*716*716*2|^0 = 835*314749909+321 by Lem1;
        then 2|^(52*4+0),321 are_congruent_mod 835 &
          321*321*321*321*2|^3 = 835*101724049+533 by LemS,Lm3;
        then 2|^(208*4+3),533 are_congruent_mod 835 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 836;
        2|^13 = 836*9+668 &
          668*668*668*668*2|^0 = 836*238176863+708 by Lm13,Lm1;
        then 2|^(13*4+0),708 are_congruent_mod 836 &
          708*708*708*708*2|^1 = 836*601113870+72 by Lem1;
        then 2|^(52*4+1),72 are_congruent_mod 836 &
          72*72*72*72*2|^0 = 836*32145+636 by LemS,Lm1;
        then 2|^(209*4+0),636 are_congruent_mod 836 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 837;
        2|^13 = 837*9+659 &
          659*659*659*659*2|^0 = 837*225328538+655 by Lm13,Lm1;
        then 2|^(13*4+0),655 are_congruent_mod 837 &
          655*655*655*655*2|^1 = 837*439814696+698 by Lem1;
        then 2|^(52*4+1),698 are_congruent_mod 837 &
          698*698*698*698*2|^1 = 837*567186947+593 by LemS;
        then 2|^(209*4+1),593 are_congruent_mod 837 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 838;
        2|^13 = 838*9+650 &
          650*650*650*650*2|^0 = 838*213014618+116 by Lm13,Lm1;
        then 2|^(13*4+0),116 are_congruent_mod 838 &
          116*116*116*116*2|^1 = 838*432133+418 by Lem1;
        then 2|^(52*4+1),418 are_congruent_mod 838 &
          418*418*418*418*2|^2 = 838*145720650+4 by LemS,Lm2;
        then 2|^(209*4+2),4 are_congruent_mod 838 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 839;
        hence thesis by XPRIMES1:839;
      end;
      suppose A: n = 840;
        2|^13 = 840*9+632 &
          632*632*632*632*2|^0 = 840*189928014+16 by Lm13,Lm1;
        then 2|^(13*4+0),16 are_congruent_mod 840 &
          16*16*16*16*2|^2 = 840*312+64 by Lem1,Lm2;
        then 2|^(52*4+2),64 are_congruent_mod 840 &
          64*64*64*64*2|^0 = 840*19972+736 by LemS,Lm1;
        then 2|^(210*4+0),736 are_congruent_mod 840 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 841;
        2|^13 = 841*9+623 &
          623*623*623*623*2|^0 = 841*179124994+687 by Lm13,Lm1;
        then 2|^(13*4+0),687 are_congruent_mod 841 &
          687*687*687*687*2|^2 = 841*1059475562+202 by Lem1,Lm2;
        then 2|^(52*4+2),202 are_congruent_mod 841 &
          202*202*202*202*2|^1 = 841*3959492+60 by LemS;
        then 2|^(210*4+1),60 are_congruent_mod 841 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 842;
        2|^13 = 842*9+614 &
          614*614*614*614*2|^0 = 842*168795705+406 by Lm13,Lm1;
        then 2|^(13*4+0),406 are_congruent_mod 842 &
          406*406*406*406*2|^2 = 842*129077942+420 by Lem1,Lm2;
        then 2|^(52*4+2),420 are_congruent_mod 842 &
          420*420*420*420*2|^2 = 842*147824038+4 by LemS;
        then 2|^(210*4+2),4 are_congruent_mod 842 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 843;
        2|^13 = 843*9+605 &
          605*605*605*605*2|^0 = 843*158925623+436 by Lm13,Lm1;
        then 2|^(13*4+0),436 are_congruent_mod 843 &
          436*436*436*436*2|^2 = 843*171466141+1 by Lem1,Lm2;
        then 2|^(52*4+2),1 are_congruent_mod 843 &
          1*1*1*1*2|^3 = 843*0+8 by LemS,Lm3;
        then 2|^(210*4+3),8 are_congruent_mod 843 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 844;
        2|^13 = 844*9+596 &
          596*596*596*596*2|^0 = 844*149500481+692 by Lm13,Lm1;
        then 2|^(13*4+0),692 are_congruent_mod 844 &
          692*692*692*692*2|^3 = 844*(15602*139313)+424 by Lem1,Lm3;
        then 2|^(52*4+3),424 are_congruent_mod 844 &
          424*424*424*424*2|^0 = 844*38293140+16 by LemS,Lm1;
        then 2|^(211*4+0),16 are_congruent_mod 844 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 845;
        2|^13 = 845*9+587 &
          587*587*587*587*2|^0 = 845*140506267+146 by Lm13,Lm1;
        then 2|^(13*4+0),146 are_congruent_mod 845 &
          146*146*146*146*2|^3 = 845*4301745+323 by Lem1,Lm3;
        then 2|^(52*4+3),323 are_congruent_mod 845 &
          323*323*323*323*2|^1 = 845*25762225+357 by LemS;
        then 2|^(211*4+1),357 are_congruent_mod 845 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 846;
        2|^13 = 846*9+578 &
          578*578*578*578*2|^0 = 846*131929218+628 by Lm13,Lm1;
        then 2|^(13*4+0),628 are_congruent_mod 846 &
          628*628*628*628*2|^3 = 846*1470815503+110 by Lem1,Lm3;
        then 2|^(52*4+3),110 are_congruent_mod 846 &
          110*110*110*110*2|^2 = 846*692245+730 by LemS,Lm2;
        then 2|^(211*4+2),730 are_congruent_mod 846 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 847;
        2|^13 = 847*9+569 &
          569*569*569*569*2|^0 = 847*123755826+499 by Lm13,Lm1;
        then 2|^(13*4+0),499 are_congruent_mod 847 &
          499*499*499*499*2|^3 = 847*585610370+618 by Lem1,Lm3;
        then 2|^(52*4+3),618 are_congruent_mod 847 &
          618*618*618*618*2|^3 = 847*1377718458+282 by LemS;
        then 2|^(211*4+3),282 are_congruent_mod 847 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 848;
        2|^13 = 848*9+560 &
          560*560*560*560*2|^1 = 848*231945660+320 by Lm13;
        then 2|^(13*4+1),320 are_congruent_mod 848 &
          320*320*320*320*2|^0 = 848*12365283+16 by Lem1,Lm1;
        then 2|^(53*4+0),16 are_congruent_mod 848 &
          16*16*16*16*2|^0 = 848*77+240 by LemS;
        then 2|^(212*4+0),240 are_congruent_mod 848 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 849;
        2|^13 = 849*9+551 &
          551*551*551*551*2|^1 = 849*217134433+785 by Lm13;
        then 2|^(13*4+1),785 are_congruent_mod 849 &
          785*785*785*785*2|^0 = 849*447271202+127 by Lem1,Lm1;
        then 2|^(53*4+0),127 are_congruent_mod 849 &
          127*127*127*127*2|^1 = 849*612826+8 by LemS;
        then 2|^(212*4+1),8 are_congruent_mod 849 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 850;
        2|^13 = 850*9+542 &
          542*542*542*542*2|^1 = 850*203052441+542 by Lm13;
        then 2|^(13*4+1),542 are_congruent_mod 850 &
          542*542*542*542*2|^0 = 850*101526220+696 by Lem1,Lm1;
        then 2|^(53*4+0),696 are_congruent_mod 850 &
          696*696*696*696*2|^2 = 850*1104276993+174 by LemS,Lm2;
        then 2|^(212*4+2),174 are_congruent_mod 850 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 851;
        2|^13 = 851*9+533 &
          533*533*533*533*2|^1 = 851*189674641+351 by Lm13;
        then 2|^(13*4+1),351 are_congruent_mod 851 &
          351*351*351*351*2|^0 = 851*17836059+192 by Lem1,Lm1;
        then 2|^(53*4+0),192 are_congruent_mod 851 &
          192*192*192*192*2|^3 = 851*12775130+338 by LemS,Lm3;
        then 2|^(212*4+3),338 are_congruent_mod 851 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 852;
        2|^13 = 852*9+524 &
          524*524*524*524*2|^1 = 852*176976478+296 by Lm13;
        then 2|^(13*4+1),296 are_congruent_mod 852 &
          296*296*296*296*2|^1 = 852*18020102+8 by Lem1;
        then 2|^(53*4+1),8 are_congruent_mod 852 &
          8*8*8*8*2|^0 = 852*4+688 by LemS,Lm1;
        then 2|^(213*4+0),688 are_congruent_mod 852 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 853;
        hence thesis by XPRIMES1:853;
      end;
      suppose A: n = 854;
        2|^13 = 854*9+506 &
          506*506*506*506*2|^1 = 854*153523262+844 by Lm13;
        then 2|^(13*4+1),844 are_congruent_mod 854 &
          844*844*844*844*2|^1 = 854*1188343271+358 by Lem1;
        then 2|^(53*4+1),358 are_congruent_mod 854 &
          358*358*358*358*2|^2 = 854*76936819+158 by LemS,Lm2;
        then 2|^(213*4+2),158 are_congruent_mod 854 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 855;
        2|^13 = 855*9+497 &
          497*497*497*497*2|^1 = 855*142721511+257 by Lm13;
        then 2|^(13*4+1),257 are_congruent_mod 855 &
          257*257*257*257*2|^1 = 855*10204609+107 by Lem1;
        then 2|^(53*4+1),107 are_congruent_mod 855 &
          107*107*107*107*2|^3 = 855*1226475+683 by LemS,Lm3;
        then 2|^(213*4+3),683 are_congruent_mod 855 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 856;
        2|^13 = 856*9+488 &
          488*488*488*488*2|^1 = 856*132505992+320 by Lm13;
        then 2|^(13*4+1),320 are_congruent_mod 856 &
          320*320*320*320*2|^2 = 856*48998878+432 by Lem1,Lm2;
        then 2|^(53*4+2),432 are_congruent_mod 856 &
          432*432*432*432*2|^0 = 856*40687520+256 by LemS,Lm1;
        then 2|^(214*4+0),256 are_congruent_mod 856 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 857;
        hence thesis by XPRIMES1:857;
      end;
      suppose A: n = 858;
        2|^13 = 858*9+470 &
          470*470*470*470*2|^1 = 858*113745477+734 by Lm13;
        then 2|^(13*4+1),734 are_congruent_mod 858 &
          734*734*734*734*2|^2 = 858*1353184277+478 by Lem1,Lm2;
        then 2|^(53*4+2),478 are_congruent_mod 858 &
          478*478*478*478*2|^2 = 858*243379665+454 by LemS;
        then 2|^(214*4+2),454 are_congruent_mod 858 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 859;
        hence thesis by XPRIMES1:859;
      end;
      suppose A: n = 860;
        2|^13 = 860*9+452 &
          452*452*452*452*2|^1 = 860*97070056+672 by Lm13;
        then 2|^(13*4+1),672 are_congruent_mod 860 &
          672*672*672*672*2|^3 = 860*1897005665+548 by Lem1,Lm3;
        then 2|^(53*4+3),548 are_congruent_mod 860 &
          548*548*548*548*2|^0 = 860*104863363+236 by LemS,Lm1;
        then 2|^(215*4+0),236 are_congruent_mod 860 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 861;
        2|^13 = 861*9+443 &
          443*443*443*443*2|^1 = 861*89462648+74 by Lm13;
        then 2|^(13*4+1),74 are_congruent_mod 861 &
          74*74*74*74*2|^3 = 861*278620+788 by Lem1,Lm3;
        then 2|^(53*4+3),788 are_congruent_mod 861 &
          788*788*788*788*2|^1 = 861*895636355+617 by LemS;
        then 2|^(215*4+1),617 are_congruent_mod 861 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 862;
        2|^13 = 862*9+434 &
          434*434*434*434*2|^1 = 862*82315505+162 by Lm13;
        then 2|^(13*4+1),162 are_congruent_mod 862 &
          162*162*162*162*2|^3 = 862*6392088+432 by Lem1,Lm3;
        then 2|^(53*4+3),432 are_congruent_mod 862 &
          432*432*432*432*2|^2 = 862*161617250+4 by LemS,Lm2;
        then 2|^(215*4+2),4 are_congruent_mod 862 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 863;
        hence thesis by XPRIMES1:863;
      end;
      suppose A: n = 864;
        2|^13 = 864*9+416 &
          416*416*416*416*2|^2 = 864*138649903+352 by Lm13,Lm2;
        then 2|^(13*4+2),352 are_congruent_mod 864 &
          352*352*352*352*2|^0 = 864*17768751+352 by Lem1,Lm1;
        then 2|^(54*4+0),352 are_congruent_mod 864 &
          352*352*352*352*2|^0 = 864*17768751+352 by LemS;
        then 2|^(216*4+0),352 are_congruent_mod 864 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 865;
        2|^13 = 865*9+407 &
          407*407*407*407*2|^2 = 865*126888283+9 by Lm13,Lm2;
        then 2|^(13*4+2),9 are_congruent_mod 865 &
          9*9*9*9*2|^0 = 865*7+506 by Lem1,Lm1;
        then 2|^(54*4+0),506 are_congruent_mod 865 &
          506*506*506*506*2|^1 = 865*151570944+32 by LemS;
        then 2|^(216*4+1),32 are_congruent_mod 865 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 866;
        2|^13 = 866*9+398 &
          398*398*398*398*2|^2 = 866*115897585+254 by Lm13,Lm2;
        then 2|^(13*4+2),254 are_congruent_mod 866 &
          254*254*254*254*2|^0 = 866*4806367+434 by Lem1,Lm1;
        then 2|^(54*4+0),434 are_congruent_mod 866 &
          434*434*434*434*2|^2 = 866*163870590+4 by LemS,Lm2;
        then 2|^(216*4+2),4 are_congruent_mod 866 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 867;
        2|^13 = 867*9+389 &
          389*389*389*389*2|^2 = 867*105642653+13 by Lm13,Lm2;
        then 2|^(13*4+2),13 are_congruent_mod 867 &
          13*13*13*13*2|^0 = 867*32+817 by Lem1,Lm1;
        then 2|^(54*4+0),817 are_congruent_mod 867 &
          817*817*817*817*2|^3 = 867*(10554*389531)+110 by LemS,Lm3;
        then 2|^(216*4+3),110 are_congruent_mod 867 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 868;
        2|^13 = 868*9+380 &
          380*380*380*380*2|^2 = 868*96089216+512 by Lm13,Lm2;
        then 2|^(13*4+2),512 are_congruent_mod 868 &
          512*512*512*512*2|^1 = 868*158339808+128 by Lem1;
        then 2|^(54*4+1),128 are_congruent_mod 868 &
          128*128*128*128*2|^0 = 868*309257+380 by LemS,Lm1;
        then 2|^(217*4+0),380 are_congruent_mod 868 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 869;
        2|^13 = 869*9+371 &
          371*371*371*371*2|^2 = 869*87203888+852 by Lm13,Lm2;
        then 2|^(13*4+2),852 are_congruent_mod 869 &
          852*852*852*852*2|^1 = 869*1212742502+194 by Lem1;
        then 2|^(54*4+1),194 are_congruent_mod 869 &
          194*194*194*194*2|^1 = 869*3259996+468 by LemS;
        then 2|^(217*4+1),468 are_congruent_mod 869 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 870;
        2|^13 = 870*9+362 &
          362*362*362*362*2|^2 = 870*78954160+544 by Lm13,Lm2;
        then 2|^(13*4+2),544 are_congruent_mod 870 &
          544*544*544*544*2|^1 = 870*201329002+452 by Lem1;
        then 2|^(54*4+1),452 are_congruent_mod 870 &
          452*452*452*452*2|^2 = 870*191908618+4 by LemS,Lm2;
        then 2|^(217*4+2),4 are_congruent_mod 870 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 871;
        2|^13 = 871*9+353 &
          353*353*353*353*2|^2 = 871*71308394+350 by Lm13,Lm2;
        then 2|^(13*4+2),350 are_congruent_mod 871 &
          350*350*350*350*2|^1 = 871*34457520+80 by Lem1;
        then 2|^(54*4+1),80 are_congruent_mod 871 &
          80*80*80*80*2|^3 = 871*376211+219 by LemS,Lm3;
        then 2|^(217*4+3),219 are_congruent_mod 871 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 872;
        2|^13 = 872*9+344 &
          344*344*344*344*2|^2 = 872*64235820+544 by Lm13,Lm2;
        then 2|^(13*4+2),544 are_congruent_mod 872 &
          544*544*544*544*2|^2 = 872*401734477+440 by Lem1;
        then 2|^(54*4+2),440 are_congruent_mod 872 &
          440*440*440*440*2|^0 = 872*42982752+256 by LemS,Lm1;
        then 2|^(218*4+0),256 are_congruent_mod 872 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 873;
        2|^13 = 873*9+335 &
          335*335*335*335*2|^2 = 873*57706532+64 by Lm13,Lm2;
        then 2|^(13*4+2),64 are_congruent_mod 873 &
          64*64*64*64*2|^2 = 873*76871+481 by Lem1;
        then 2|^(54*4+2),481 are_congruent_mod 873 &
          481*481*481*481*2|^1 = 873*122629810+512 by LemS;
        then 2|^(218*4+1),512 are_congruent_mod 873 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 874;
        2|^13 = 874*9+326 &
          326*326*326*326*2|^2 = 874*51691479+58 by Lm13,Lm2;
        then 2|^(13*4+2),58 are_congruent_mod 874 &
          58*58*58*58*2|^2 = 874*51791+650 by Lem1;
        then 2|^(54*4+2),650 are_congruent_mod 874 &
          650*650*650*650*2|^2 = 874*816962242+492 by LemS;
        then 2|^(218*4+2),492 are_congruent_mod 874 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 875;
        2|^13 = 875*9+317 &
          317*317*317*317*2|^2 = 875*46162464+484 by Lm13,Lm2;
        then 2|^(13*4+2),484 are_congruent_mod 875 &
          484*484*484*484*2|^2 = 875*250861136+144 by Lem1;
        then 2|^(54*4+2),144 are_congruent_mod 875 &
          144*144*144*144*2|^3 = 875*3931261+193 by LemS,Lm3;
        then 2|^(218*4+3),193 are_congruent_mod 875 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 876;
        2|^13 = 876*9+308 &
          308*308*308*308*2|^2 = 876*41092139+220 by Lm13,Lm2;
        then 2|^(13*4+2),220 are_congruent_mod 876 &
          220*220*220*220*2|^3 = 876*21393242+8 by Lem1,Lm3;
        then 2|^(54*4+3),8 are_congruent_mod 876 &
          8*8*8*8*2|^0 = 876*4+592 by LemS,Lm1;
        then 2|^(219*4+0),592 are_congruent_mod 876 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 877;
        hence thesis by XPRIMES1:877;
      end;
      suppose A: n = 878;
        2|^13 = 878*9+290 &
          290*290*290*290*2|^2 = 878*32222369+18 by Lm13,Lm2;
        then 2|^(13*4+2),18 are_congruent_mod 878 &
          18*18*18*18*2|^3 = 878*956+440 by Lem1,Lm3;
        then 2|^(54*4+3),440 are_congruent_mod 878 &
          440*440*440*440*2|^2 = 878*170756082+4 by LemS,Lm2;
        then 2|^(219*4+2),4 are_congruent_mod 878 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 879;
        2|^13 = 879*9+281 &
          281*281*281*281*2|^2 = 879*28372421+25 by Lm13,Lm2;
        then 2|^(13*4+2),25 are_congruent_mod 879 &
          25*25*25*25*2|^3 = 879*3555+155 by Lem1,Lm3;
        then 2|^(54*4+3),155 are_congruent_mod 879 &
          155*155*155*155*2|^3 = 879*5253248+8 by LemS;
        then 2|^(219*4+3),8 are_congruent_mod 879 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 880;
        2|^13 = 880*9+272 &
          272*272*272*272*2|^3 = 880*49760293+208 by Lm13,Lm3;
        then 2|^(13*4+3),208 are_congruent_mod 880 &
          208*208*208*208*2|^0 = 880*2127015+496 by Lem1,Lm1;
        then 2|^(55*4+0),496 are_congruent_mod 880 &
          496*496*496*496*2|^0 = 880*68777127+496 by LemS;
        then 2|^(220*4+0),496 are_congruent_mod 880 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 881;
        hence thesis by XPRIMES1:881;
      end;
      suppose A: n = 882;
        2|^13 = 882*9+254 &
          254*254*254*254*2|^3 = 882*37753417+254 by Lm13,Lm3;
        then 2|^(13*4+3),254 are_congruent_mod 882 &
          254*254*254*254*2|^0 = 882*4719177+142 by Lem1,Lm1;
        then 2|^(55*4+0),142 are_congruent_mod 882 &
          142*142*142*142*2|^2 = 882*1843931+442 by LemS,Lm2;
        then 2|^(220*4+2),442 are_congruent_mod 882 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 883;
        hence thesis by XPRIMES1:883;
      end;
      suppose A: n = 884;
        2|^13 = 884*9+236 &
          236*236*236*236*2|^3 = 884*28072800+128 by Lm13,Lm3;
        then 2|^(13*4+3),128 are_congruent_mod 884 &
          128*128*128*128*2|^1 = 884*607320+32 by Lem1;
        then 2|^(55*4+1),32 are_congruent_mod 884 &
          32*32*32*32*2|^0 = 884*1186+152 by LemS,Lm1;
        then 2|^(221*4+0),152 are_congruent_mod 884 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 885;
        2|^13 = 885*9+227 &
          227*227*227*227*2|^3 = 885*24002149+863 by Lm13,Lm3;
        then 2|^(13*4+3),863 are_congruent_mod 885 &
          863*863*863*863*2|^1 = 885*1253516075+347 by Lem1;
        then 2|^(55*4+1),347 are_congruent_mod 885 &
          347*347*347*347*2|^1 = 885*32764581+377 by LemS;
        then 2|^(221*4+1),377 are_congruent_mod 885 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 886;
        2|^13 = 886*9+218 &
          218*218*218*218*2|^3 = 886*20393052+536 by Lm13,Lm3;
        then 2|^(13*4+3),536 are_congruent_mod 886 &
          536*536*536*536*2|^1 = 886*186318265+442 by Lem1;
        then 2|^(55*4+1),442 are_congruent_mod 886 &
          442*442*442*442*2|^2 = 886*172311930+4 by LemS,Lm2;
        then 2|^(221*4+2),4 are_congruent_mod 886 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 887;
        hence thesis by XPRIMES1:887;
      end;
      suppose A: n = 888;
        2|^13 = 888*9+200 &
          200*200*200*200*2|^3 = 888*14414414+368 by Lm13,Lm3;
        then 2|^(13*4+3),368 are_congruent_mod 888 &
          368*368*368*368*2|^2 = 888*82611080+64 by Lem1,Lm2;
        then 2|^(55*4+2),64 are_congruent_mod 888 &
          64*64*64*64*2|^0 = 888*18893+232 by LemS,Lm1;
        then 2|^(222*4+0),232 are_congruent_mod 888 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 889;
        2|^13 = 889*9+191 &
          191*191*191*191*2|^3 = 889*11976273+191 by Lm13,Lm3;
        then 2|^(13*4+3),191 are_congruent_mod 889 &
          191*191*191*191*2|^2 = 889*5988136+540 by Lem1,Lm2;
        then 2|^(55*4+2),540 are_congruent_mod 889 &
          540*540*540*540*2|^1 = 889*191294848+128 by LemS;
        then 2|^(222*4+1),128 are_congruent_mod 889 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 890;
        2|^13 = 890*9+182 &
          182*182*182*182*2|^3 = 890*9862466+268 by Lm13,Lm3;
        then 2|^(13*4+3),268 are_congruent_mod 890 &
          268*268*268*268*2|^2 = 890*23185110+4 by Lem1,Lm2;
        then 2|^(55*4+2),4 are_congruent_mod 890 &
          4*4*4*4*2|^2 = 890*1+134 by LemS;
        then 2|^(222*4+2),134 are_congruent_mod 890 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 891;
        2|^13 = 891*9+173 &
          173*173*173*173*2|^3 = 891*8042604+164 by Lm13,Lm3;
        then 2|^(13*4+3),164 are_congruent_mod 891 &
          164*164*164*164*2|^2 = 891*3247563+631 by Lem1,Lm2;
        then 2|^(55*4+2),631 are_congruent_mod 891 &
          631*631*631*631*2|^3 = 891*1423409040+728 by LemS,Lm3;
        then 2|^(222*4+3),728 are_congruent_mod 891 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 892;
        2|^13 = 892*9+164 &
          164*164*164*164*2|^3 = 892*6487845+788 by Lm13,Lm3;
        then 2|^(13*4+3),788 are_congruent_mod 892 &
          788*788*788*788*2|^3 = 892*(48953*70640)+448 by Lem1;
        then 2|^(55*4+3),448 are_congruent_mod 892 &
          448*448*448*448*2|^0 = 892*45159300+16 by LemS,Lm1;
        then 2|^(223*4+0),16 are_congruent_mod 892 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 893;
        2|^13 = 893*9+155 &
          155*155*155*155*2|^3 = 893*5170890+230 by Lm13,Lm3;
        then 2|^(13*4+3),230 are_congruent_mod 893 &
          230*230*230*230*2|^3 = 893*25069742+394 by Lem1;
        then 2|^(55*4+3),394 are_congruent_mod 893 &
          394*394*394*394*2|^1 = 893*53971367+661 by LemS;
        then 2|^(223*4+1),661 are_congruent_mod 893 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 894;
        2|^13 = 894*9+146 &
          146*146*146*146*2|^3 = 894*4065967+350 by Lm13,Lm3;
        then 2|^(13*4+3),350 are_congruent_mod 894 &
          350*350*350*350*2|^3 = 894*134284116+296 by Lem1;
        then 2|^(55*4+3),296 are_congruent_mod 894 &
          296*296*296*296*2|^2 = 894*34347040+64 by LemS,Lm2;
        then 2|^(223*4+2),64 are_congruent_mod 894 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 895;
        2|^13 = 895*9+137 & 137*137*137*137*2|^3 = 895*3148830+38 by Lm13,Lm3;
        then 2|^(13*4+3),38 are_congruent_mod 895 &
          38*38*38*38*2|^3 = 895*18638+78 by Lem1;
        then 2|^(55*4+3),78 are_congruent_mod 895 &
          78*78*78*78*2|^3 = 895*330860+748 by LemS;
        then 2|^(223*4+3),748 are_congruent_mod 895 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 896;
        2|^14 = 896*18+256 &
          256*256*256*256*2|^0 = 896*4793490+256 by Lm14,Lm1;
        then 2|^(14*4+0),256 are_congruent_mod 896 &
          256*256*256*256*2|^0 = 896*4793490+256 by Lem1;
        then 2|^(56*4+0),256 are_congruent_mod 896 &
          256*256*256*256*2|^0 = 896*4793490+256 by LemS;
        then 2|^(224*4+0),256 are_congruent_mod 896 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 897;
        2|^14 = 897*18+238 &
          238*238*238*238*2|^0 = 897*3576970+646 by Lm14,Lm1;
        then 2|^(14*4+0),646 are_congruent_mod 897 &
          646*646*646*646*2|^0 = 897*194150104+568 by Lem1;
        then 2|^(56*4+0),568 are_congruent_mod 897 &
          568*568*568*568*2|^1 = 897*232076355+317 by LemS;
        then 2|^(224*4+1),317 are_congruent_mod 897 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 898;
        2|^14 = 898*18+220 &
          220*220*220*220*2|^0 = 898*2608641+382 by Lm14,Lm1;
        then 2|^(14*4+0),382 are_congruent_mod 898 &
          382*382*382*382*2|^0 = 898*23712487+450 by Lem1;
        then 2|^(56*4+0),450 are_congruent_mod 898 &
          450*450*450*450*2|^2 = 898*182655902+4 by LemS,Lm2;
        then 2|^(224*4+2),4 are_congruent_mod 898 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 899;
        2|^14 = 899*18+202 &
          202*202*202*202*2|^0 = 899*1852020+436 by Lm14,Lm1;
        then 2|^(14*4+0),436 are_congruent_mod 899 &
          436*436*436*436*2|^0 = 899*40196317+233 by Lem1;
        then 2|^(56*4+0),233 are_congruent_mod 899 &
          233*233*233*233*2|^3 = 899*26227323+791 by LemS,Lm3;
        then 2|^(224*4+3),791 are_congruent_mod 899 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 900;
        2|^14 = 900*18+184 &
          184*184*184*184*2|^0 = 900*1273587+436 by Lm14,Lm1;
        then 2|^(14*4+0),436 are_congruent_mod 900 &
          436*436*436*436*2|^1 = 900*80303309+332 by Lem1;
        then 2|^(56*4+1),332 are_congruent_mod 900 &
          332*332*332*332*2|^0 = 900*13499255+676 by LemS,Lm1;
        then 2|^(225*4+0),676 are_congruent_mod 900 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 901;
        2|^14 = 901*18+166 & 166*166*166*166*2|^0 = 901*842767+69 by Lm14,Lm1;
        then 2|^(14*4+0),69 are_congruent_mod 901 &
          69*69*69*69*2|^1 = 901*50315+427 by Lem1;
        then 2|^(56*4+1),427 are_congruent_mod 901 &
          427*427*427*427*2|^1 = 901*73793261+321 by LemS;
        then 2|^(225*4+1),321 are_congruent_mod 901 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 902;
        2|^14 = 902*18+148 &
          148*148*148*148*2|^0 = 902*531912+592 by Lm14,Lm1;
        then 2|^(14*4+0),592 are_congruent_mod 902 &
          592*592*592*592*2|^1 = 902*272339280+32 by Lem1;
        then 2|^(56*4+1),32 are_congruent_mod 902 &
          32*32*32*32*2|^2 = 902*4650+4 by LemS,Lm2;
        then 2|^(225*4+2),4 are_congruent_mod 902 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 903;
        2|^14 = 903*18+130 &
          130*130*130*130*2|^0 = 903*316290+130 by Lm14,Lm1;
        then 2|^(14*4+0),130 are_congruent_mod 903 &
          130*130*130*130*2|^1 = 903*632580+260 by Lem1;
        then 2|^(56*4+1),260 are_congruent_mod 903 &
          260*260*260*260*2|^3 = 903*40485138+386 by LemS,Lm3;
        then 2|^(225*4+3),386 are_congruent_mod 903 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 904;
        2|^14 = 904*18+112 &
          112*112*112*112*2|^0 = 904*174061+792 by Lm14,Lm1;
        then 2|^(14*4+0),792 are_congruent_mod 904 &
          792*792*792*792*2|^2 = 904*1740974007+456 by Lem1,Lm2;
        then 2|^(56*4+2),456 are_congruent_mod 904 &
          456*456*456*456*2|^0 = 904*47828960+256 by LemS,Lm1;
        then 2|^(226*4+0),256 are_congruent_mod 904 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 904 < n & n <= 1004;
      then 904+1 <= n & n <= 1004 by NAT_1:13;
      then n = 905 or ... or n = 1004;
      then per cases;
      suppose A: n = 905;
        2|^14 = 905*18+94 & 94*94*94*94*2|^0 = 905*86270+546 by Lm14,Lm1;
        then 2|^(14*4+0),546 are_congruent_mod 905 &
          546*546*546*546*2|^2 = 905*392809500+324 by Lem1,Lm2;
        then 2|^(56*4+2),324 are_congruent_mod 905 &
          324*324*324*324*2|^1 = 905*24353504+32 by LemS;
        then 2|^(226*4+1),32 are_congruent_mod 905 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 906;
        2|^14 = 906*18+76 & 76*76*76*76*2|^0 = 906*36823+538 by Lm14,Lm1;
        then 2|^(14*4+0),538 are_congruent_mod 906 &
          538*538*538*538*2|^2 = 906*369880040+304 by Lem1,Lm2;
        then 2|^(56*4+2),304 are_congruent_mod 906 &
          304*304*304*304*2|^2 = 906*37707360+64 by LemS;
        then 2|^(226*4+2),64 are_congruent_mod 906 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 907;
        hence thesis by XPRIMES1:907;
      end;
      suppose A: n = 908;
        2|^14 = 908*18+40 & 40*40*40*40*2|^0 = 908*2819+348 by Lm14,Lm1;
        then 2|^(14*4+0),348 are_congruent_mod 908 &
          348*348*348*348*2|^3 = 908*129217434+456 by Lem1,Lm3;
        then 2|^(56*4+3),456 are_congruent_mod 908 &
          456*456*456*456*2|^0 = 908*47618260+16 by LemS,Lm1;
        then 2|^(227*4+0),16 are_congruent_mod 908 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 909;
        2|^14 = 909*18+22 & 22*22*22*22*2|^0 = 909*257+643 by Lm14,Lm1;
        then 2|^(14*4+0),643 are_congruent_mod 909 &
          643*643*643*643*2|^3 = 909*1504423107+545 by Lem1,Lm3;
        then 2|^(56*4+3),545 are_congruent_mod 909 &
          545*545*545*545*2|^1 = 909*194111882+512 by LemS;
        then 2|^(227*4+1),512 are_congruent_mod 909 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 910;
        2|^14 = 910*18+4 & 4*4*4*4*2|^0 = 910*0+256 by Lm14,Lm1;
        then 2|^(14*4+0),256 are_congruent_mod 910 &
          256*256*256*256*2|^3 = 910*37757954+228 by Lem1,Lm3;
        then 2|^(56*4+3),228 are_congruent_mod 910 &
          228*228*228*228*2|^2 = 910*11878401+114 by LemS,Lm2;
        then 2|^(227*4+2),114 are_congruent_mod 910 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 911;
        hence thesis by XPRIMES1:911;
      end;
      suppose A: n = 912;
        2|^14 = 912*17+880 &
          880*880*880*880*2|^1 = 912*1315121403+464 by Lm14;
        then 2|^(14*4+1),464 are_congruent_mod 912 &
          464*464*464*464*2|^0 = 912*50824964+448 by Lem1,Lm1;
        then 2|^(57*4+0),448 are_congruent_mod 912 &
          448*448*448*448*2|^0 = 912*44168964+448 by LemS;
        then 2|^(228*4+0),448 are_congruent_mod 912 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 913;
        2|^14 = 913*17+863 &
          863*863*863*863*2|^1 = 913*1215073085+117 by Lm14;
        then 2|^(14*4+1),117 are_congruent_mod 913 &
          117*117*117*117*2|^0 = 913*205245+36 by Lem1,Lm1;
        then 2|^(57*4+0),36 are_congruent_mod 913 &
          36*36*36*36*2|^1 = 913*3679+305 by LemS;
        then 2|^(228*4+1),305 are_congruent_mod 913 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 914;
        2|^14 = 914*17+846 &
          846*846*846*846*2|^1 = 914*1120895826+348 by Lm14;
        then 2|^(14*4+1),348 are_congruent_mod 914 &
          348*348*348*348*2|^0 = 914*16046147+458 by Lem1,Lm1;
        then 2|^(57*4+0),458 are_congruent_mod 914 &
          458*458*458*458*2|^2 = 914*192564270+4 by LemS,Lm2;
        then 2|^(228*4+2),4 are_congruent_mod 914 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 915;
        2|^14 = 915*17+829 &
          829*829*829*829*2|^1 = 915*1032350146+572 by Lm14;
        then 2|^(14*4+1),572 are_congruent_mod 915 &
          572*572*572*572*2|^0 = 915*116993846+766 by Lem1,Lm1;
        then 2|^(57*4+0),766 are_congruent_mod 915 &
          766*766*766*766*2|^3 = 915*(223*13498301)+743 by LemS,Lm3;
        then 2|^(228*4+3),743 are_congruent_mod 915 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 916;
        2|^14 = 916*17+812 &
          812*812*812*812*2|^1 = 916*949201987+580 by Lm14;
        then 2|^(14*4+1),580 are_congruent_mod 916 &
          580*580*580*580*2|^1 = 916*247085065+460 by Lem1;
        then 2|^(57*4+1),460 are_congruent_mod 916 &
          460*460*460*460*2|^0 = 916*48880524+16 by LemS,Lm1;
        then 2|^(229*4+0),16 are_congruent_mod 916 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 917;
        2|^14 = 917*17+795 &
          795*795*795*795*2|^1 = 917*871222684+22 by Lm14;
        then 2|^(14*4+1),22 are_congruent_mod 917 &
          22*22*22*22*2|^1 = 917*510+842 by Lem1;
        then 2|^(57*4+1),842 are_congruent_mod 917 &
          842*842*842*842*2|^1 = 917*1096248534+914 by LemS;
        then 2|^(229*4+1),914 are_congruent_mod 917 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 918;
        2|^14 = 918*17+778 &
          778*778*778*778*2|^1 = 918*798188933+818 by Lm14;
        then 2|^(14*4+1),818 are_congruent_mod 918 &
          818*818*818*818*2|^1 = 918*975439928+848 by Lem1;
        then 2|^(57*4+1),848 are_congruent_mod 918 &
          848*848*848*848*2|^2 = 918*(42301*53266)+676 by LemS,Lm2;
        then 2|^(229*4+2),676 are_congruent_mod 918 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 919;
        hence thesis by XPRIMES1:919;
      end;
      suppose A: n = 920;
        2|^14 = 920*17+744 &
          744*744*744*744*2|^1 = 920*666091528+832 by Lm14;
        then 2|^(14*4+1),832 are_congruent_mod 920 &
          832*832*832*832*2|^2 = 920*2083365505+104 by Lem1,Lm2;
        then 2|^(57*4+2),104 are_congruent_mod 920 &
          104*104*104*104*2|^0 = 920*127158+496 by LemS,Lm1;
        then 2|^(230*4+0),496 are_congruent_mod 920 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 921;
        2|^14 = 921*17+727 &
          727*727*727*727*2|^1 = 921*606607825+857 by Lm14;
        then 2|^(14*4+1),857 are_congruent_mod 921 &
          857*857*857*857*2|^2 = 921*(195*12014039)+199 by Lem1,Lm2;
        then 2|^(57*4+2),199 are_congruent_mod 921 &
          199*199*199*199*2|^1 = 921*3405514+8 by LemS;
        then 2|^(230*4+1),8 are_congruent_mod 921 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 922;
        2|^14 = 922*17+710 &
          710*710*710*710*2|^1 = 922*551229522+716 by Lm14;
        then 2|^(14*4+1),716 are_congruent_mod 922 &
          716*716*716*716*2|^2 = 922*1140200322+460 by Lem1,Lm2;
        then 2|^(57*4+2),460 are_congruent_mod 922 &
          460*460*460*460*2|^2 = 922*194249718+4 by LemS;
        then 2|^(230*4+2),4 are_congruent_mod 922 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 923;
        2|^14 = 923*17+693 &
          693*693*693*693*2|^1 = 923*499759700+902 by Lm14;
        then 2|^(14*4+1),902 are_congruent_mod 923 &
          902*902*902*902*2|^2 = 923*(62*46269281)+758 by Lem1,Lm2;
        then 2|^(57*4+2),758 are_congruent_mod 923 &
          758*758*758*758*2|^3 = 923*(39757*71970)+98 by LemS,Lm3;
        then 2|^(230*4+3),98 are_congruent_mod 923 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 924;
        2|^14 = 924*17+676 &
          676*676*676*676*2|^1 = 924*452006633+260 by Lm14;
        then 2|^(14*4+1),260 are_congruent_mod 924 &
          260*260*260*260*2|^3 = 924*39565021+596 by Lem1,Lm3;
        then 2|^(57*4+3),596 are_congruent_mod 924 &
          596*596*596*596*2|^0 = 924*136556717+148 by LemS,Lm1;
        then 2|^(231*4+0),148 are_congruent_mod 924 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 925;
        2|^14 = 925*17+659 &
          659*659*659*659*2|^1 = 925*407783755+547 by Lm14;
        then 2|^(14*4+1),547 are_congruent_mod 925 &
          547*547*547*547*2|^3 = 925*774279141+23 by Lem1,Lm3;
        then 2|^(57*4+3),23 are_congruent_mod 925 &
          23*23*23*23*2|^1 = 925*605+57 by LemS;
        then 2|^(231*4+1),57 are_congruent_mod 925 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 926;
        2|^14 = 926*17+642 &
          642*642*642*642*2|^1 = 926*366909639+78 by Lm14;
        then 2|^(14*4+1),78 are_congruent_mod 926 &
          78*78*78*78*2|^3 = 926*319784+464 by Lem1,Lm3;
        then 2|^(57*4+3),464 are_congruent_mod 926 &
          464*464*464*464*2|^2 = 926*200226210+4 by LemS,Lm2;
        then 2|^(231*4+2),4 are_congruent_mod 926 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 927;
        2|^14 = 927*17+625 &
          625*625*625*625*2|^1 = 927*329207962+476 by Lm14;
        then 2|^(14*4+1),476 are_congruent_mod 927 &
          476*476*476*476*2|^3 = 927*443035027+179 by Lem1,Lm3;
        then 2|^(57*4+3),179 are_congruent_mod 927 &
          179*179*179*179*2|^3 = 927*8859768+512 by LemS;
        then 2|^(231*4+3),512 are_congruent_mod 927 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 928;
        2|^14 = 928*17+608 &
          608*608*608*608*2|^2 = 928*589014969+352 by Lm14,Lm2;
        then 2|^(14*4+2),352 are_congruent_mod 928 &
          352*352*352*352*2|^0 = 928*16543320+256 by Lem1,Lm1;
        then 2|^(58*4+0),256 are_congruent_mod 928 &
          256*256*256*256*2|^0 = 928*4628197+480 by LemS;
        then 2|^(232*4+0),480 are_congruent_mod 928 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 929;
        hence thesis by XPRIMES1:929;
      end;
      suppose A: n = 930;
        2|^14 = 930*17+574 &
          574*574*574*574*2|^2 = 930*466900793+814 by Lm14,Lm2;
        then 2|^(14*4+2),814 are_congruent_mod 930 &
          814*814*814*814*2|^0 = 930*472078988+376 by Lem1,Lm1;
        then 2|^(58*4+0),376 are_congruent_mod 930 &
          376*376*376*376*2|^2 = 930*85966337+94 by LemS,Lm2;
        then 2|^(232*4+2),94 are_congruent_mod 930 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 931;
        2|^14 = 931*17+557 &
          557*557*557*557*2|^2 = 931*413552919+415 by Lm14,Lm2;
        then 2|^(14*4+2),415 are_congruent_mod 931 &
          415*415*415*415*2|^0 = 931*31859775+100 by Lem1,Lm1;
        then 2|^(58*4+0),100 are_congruent_mod 931 &
          100*100*100*100*2|^3 = 931*859291+79 by LemS,Lm3;
        then 2|^(232*4+3),79 are_congruent_mod 931 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 932;
        2|^14 = 932*17+540 &
          540*540*540*540*2|^2 = 932*364938025+700 by Lm14,Lm2;
        then 2|^(14*4+2),700 are_congruent_mod 932 &
          700*700*700*700*2|^1 = 932*515236051+468 by Lem1;
        then 2|^(58*4+1),468 are_congruent_mod 932 &
          468*468*468*468*2|^0 = 932*51471580+16 by LemS,Lm1;
        then 2|^(233*4+0),16 are_congruent_mod 932 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 933;
        2|^14 = 933*17+523 &
          523*523*523*523*2|^2 = 933*320763617+703 by Lm14,Lm2;
        then 2|^(14*4+2),703 are_congruent_mod 933 &
          703*703*703*703*2|^1 = 933*523563849+245 by Lem1;
        then 2|^(58*4+1),245 are_congruent_mod 933 &
          245*245*245*245*2|^1 = 933*7723474+8 by LemS;
        then 2|^(233*4+1),8 are_congruent_mod 933 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 934;
        2|^14 = 934*17+506 &
          506*506*506*506*2|^2 = 934*280747037+626 by Lm14,Lm2;
        then 2|^(14*4+2),626 are_congruent_mod 934 &
          626*626*626*626*2|^1 = 934*328836829+466 by Lem1;
        then 2|^(58*4+1),466 are_congruent_mod 934 &
          466*466*466*466*2|^2 = 934*201956010+4 by LemS,Lm2;
        then 2|^(233*4+2),4 are_congruent_mod 934 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 935;
        2|^14 = 935*17+489 &
          489*489*489*489*2|^2 = 935*244615412+344 by Lm14,Lm2;
        then 2|^(14*4+2),344 are_congruent_mod 935 &
          344*344*344*344*2|^1 = 935*29953815+767 by Lem1;
        then 2|^(58*4+1),767 are_congruent_mod 935 &
          767*767*767*767*2|^3 = 935*(37925*78079)+43 by LemS,Lm3;
        then 2|^(233*4+3),43 are_congruent_mod 935 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 936;
        2|^14 = 936*17+472 &
          472*472*472*472*2|^2 = 936*212105601+88 by Lm14,Lm2;
        then 2|^(14*4+2),88 are_congruent_mod 936 &
          88*88*88*88*2|^2 = 936*256280+64 by Lem1;
        then 2|^(58*4+2),64 are_congruent_mod 936 &
          64*64*64*64*2|^0 = 936*17924+352 by LemS,Lm1;
        then 2|^(234*4+0),352 are_congruent_mod 936 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 937;
        hence thesis by XPRIMES1:937;
      end;
      suppose A: n = 938;
        2|^14 = 938*17+438 &
          438*438*438*438*2|^2 = 938*156947208+240 by Lm14,Lm2;
        then 2|^(14*4+2),240 are_congruent_mod 938 &
          240*240*240*240*2|^2 = 938*14148230+260 by Lem1;
        then 2|^(58*4+2),260 are_congruent_mod 938 &
          260*260*260*260*2|^2 = 938*19487249+438 by LemS;
        then 2|^(234*4+2),438 are_congruent_mod 938 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 939;
        2|^14 = 939*17+421 &
          421*421*421*421*2|^2 = 939*133820541+325 by Lm14,Lm2;
        then 2|^(14*4+2),325 are_congruent_mod 939 &
          325*325*325*325*2|^2 = 939*47525625+625 by Lem1;
        then 2|^(58*4+2),625 are_congruent_mod 939 &
          625*625*625*625*2|^3 = 939*1300003328+8 by LemS,Lm3;
        then 2|^(234*4+3),8 are_congruent_mod 939 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 940;
        2|^14 = 940*17+404 &
          404*404*404*404*2|^2 = 940*113359415+524 by Lm14,Lm2;
        then 2|^(14*4+2),524 are_congruent_mod 940 &
          524*524*524*524*2|^3 = 940*641633870+408 by Lem1,Lm3;
        then 2|^(58*4+3),408 are_congruent_mod 940 &
          408*408*408*408*2|^0 = 940*29479003+476 by LemS,Lm1;
        then 2|^(235*4+0),476 are_congruent_mod 940 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 941;
        hence thesis by XPRIMES1:941;
      end;
      suppose A: n = 942;
        2|^14 = 942*17+370 &
          370*370*370*370*2|^2 = 942*79582208+64 by Lm14,Lm2;
        then 2|^(14*4+2),64 are_congruent_mod 942 &
          64*64*64*64*2|^3 = 942*142481+626 by Lem1,Lm3;
        then 2|^(58*4+3),626 are_congruent_mod 942 &
          626*626*626*626*2|^2 = 942*652088320+64 by LemS,Lm2;
        then 2|^(235*4+2),64 are_congruent_mod 942 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 943;
        2|^14 = 943*17+353 &
          353*353*353*353*2|^2 = 943*65863851+31 by Lm14,Lm2;
        then 2|^(14*4+2),31 are_congruent_mod 943 &
          31*31*31*31*2|^3 = 943*7834+706 by Lem1,Lm3;
        then 2|^(58*4+3),706 are_congruent_mod 943 &
          706*706*706*706*2|^3 = 943*2107643233+49 by LemS;
        then 2|^(235*4+3),49 are_congruent_mod 943 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 944;
        2|^14 = 944*17+336 &
          336*336*336*336*2|^3 = 944*108012769+592 by Lm14,Lm3;
        then 2|^(14*4+3),592 are_congruent_mod 944 &
          592*592*592*592*2|^0 = 944*130111245+16 by Lem1,Lm1;
        then 2|^(59*4+0),16 are_congruent_mod 944 &
          16*16*16*16*2|^0 = 944*69+400 by LemS;
        then 2|^(236*4+0),400 are_congruent_mod 944 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 945;
        2|^14 = 945*17+319 &
          319*319*319*319*2|^3 = 945*87663924+788 by Lm14,Lm3;
        then 2|^(14*4+3),788 are_congruent_mod 945 &
          788*788*788*788*2|^0 = 945*408012117+571 by Lem1,Lm1;
        then 2|^(59*4+0),571 are_congruent_mod 945 &
          571*571*571*571*2|^1 = 945*224979330+512 by LemS;
        then 2|^(236*4+1),512 are_congruent_mod 945 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 946;
        2|^14 = 946*17+302 &
          302*302*302*302*2|^3 = 946*70343929+94 by Lm14,Lm3;
        then 2|^(14*4+3),94 are_congruent_mod 946 &
          94*94*94*94*2|^0 = 946*82531+570 by Lem1,Lm1;
        then 2|^(59*4+0),570 are_congruent_mod 946 &
          570*570*570*570*2|^2 = 946*446342536+944 by LemS,Lm2;
        then 2|^(236*4+2),944 are_congruent_mod 946 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 947;
        hence thesis by XPRIMES1:947;
      end;
      suppose A: n = 948;
        2|^14 = 948*17+268 &
          268*268*268*268*2|^3 = 948*43533223+404 by Lm14,Lm3;
        then 2|^(14*4+3),404 are_congruent_mod 948 &
          404*404*404*404*2|^1 = 948*56201398+8 by Lem1;
        then 2|^(59*4+1),8 are_congruent_mod 948 &
          8*8*8*8*2|^0 = 948*4+304 by LemS,Lm1;
        then 2|^(237*4+0),304 are_congruent_mod 948 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 949;
        2|^14 = 949*17+251 &
          251*251*251*251*2|^3 = 949*33459439+397 by Lm14,Lm3;
        then 2|^(14*4+3),397 are_congruent_mod 949 &
          397*397*397*397*2|^1 = 949*52351099+811 by Lem1;
        then 2|^(59*4+1),811 are_congruent_mod 949 &
          811*811*811*811*2|^1 = 949*911690018+600 by LemS;
        then 2|^(237*4+1),600 are_congruent_mod 949 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 950;
        2|^14 = 950*17+234 &
          234*234*234*234*2|^3 = 950*25248164+488 by Lm14,Lm3;
        then 2|^(14*4+3),488 are_congruent_mod 950 &
          488*488*488*488*2|^1 = 950*119394873+122 by Lem1;
        then 2|^(59*4+1),122 are_congruent_mod 950 &
          122*122*122*122*2|^2 = 950*932772+424 by LemS,Lm2;
        then 2|^(237*4+2),424 are_congruent_mod 950 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 951;
        2|^14 = 951*17+217 &
          217*217*217*217*2|^3 = 951*18652987+731 by Lm14,Lm3;
        then 2|^(14*4+3),731 are_congruent_mod 951 &
          731*731*731*731*2|^1 = 951*600508261+431 by Lem1;
        then 2|^(59*4+1),431 are_congruent_mod 951 &
          431*431*431*431*2|^3 = 951*290280960+8 by LemS,Lm3;
        then 2|^(237*4+3),8 are_congruent_mod 951 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 952;
        2|^14 = 952*17+200 &
          200*200*200*200*2|^3 = 952*13445378+144 by Lm14,Lm3;
        then 2|^(14*4+3),144 are_congruent_mod 952 &
          144*144*144*144*2|^2 = 952*1806645+744 by Lem1,Lm2;
        then 2|^(59*4+2),744 are_congruent_mod 952 &
          744*744*744*744*2|^0 = 952*321850948+800 by LemS,Lm1;
        then 2|^(238*4+0),800 are_congruent_mod 952 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 953;
        hence thesis by XPRIMES1:953;
      end;
      suppose A: n = 954;
        2|^14 = 954*17+166 &
          166*166*166*166*2|^3 = 954*6367573+446 by Lm14,Lm3;
        then 2|^(14*4+3),446 are_congruent_mod 954 &
          446*446*446*446*2|^2 = 954*165901782+196 by Lem1,Lm2;
        then 2|^(59*4+2),196 are_congruent_mod 954 &
          196*196*196*196*2|^2 = 954*6187794+748 by LemS;
        then 2|^(238*4+2),748 are_congruent_mod 954 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 955;
        2|^14 = 955*17+149 &
          149*149*149*149*2|^3 = 955*4128874+538 by Lm14,Lm3;
        then 2|^(14*4+3),538 are_congruent_mod 955 &
          538*538*538*538*2|^2 = 955*350901902+134 by Lem1,Lm2;
        then 2|^(59*4+2),134 are_congruent_mod 955 &
          134*134*134*134*2|^3 = 955*2700883+223 by LemS,Lm3;
        then 2|^(238*4+3),223 are_congruent_mod 955 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 956;
        2|^14 = 956*17+132 &
          132*132*132*132*2|^3 = 956*2540550+408 by Lm14,Lm3;
        then 2|^(14*4+3),408 are_congruent_mod 956 &
          408*408*408*408*2|^3 = 956*231885048+480 by Lem1;
        then 2|^(59*4+3),480 are_congruent_mod 956 &
          480*480*480*480*2|^0 = 956*55527364+16 by LemS,Lm1;
        then 2|^(239*4+0),16 are_congruent_mod 956 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 957;
        2|^14 = 957*17+115 &
          115*115*115*115*2|^3 = 957*1462074+182 by Lm14,Lm3;
        then 2|^(14*4+3),182 are_congruent_mod 957 &
          182*182*182*182*2|^3 = 957*9171990+578 by Lem1;
        then 2|^(59*4+3),578 are_congruent_mod 957 &
          578*578*578*578*2|^1 = 957*233254167+293 by LemS;
        then 2|^(239*4+1),293 are_congruent_mod 957 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 958;
        2|^14 = 958*17+98 & 98*98*98*98*2|^3 = 958*770244+776 by Lm14,Lm3;
        then 2|^(14*4+3),776 are_congruent_mod 958 &
          776*776*776*776*2|^3 = 958*(16*189256751)+480 by Lem1;
        then 2|^(59*4+3),480 are_congruent_mod 958 &
          480*480*480*480*2|^2 = 958*221645762+4 by LemS,Lm2;
        then 2|^(239*4+2),4 are_congruent_mod 958 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 959;
        2|^14 = 959*17+81 & 81*81*81*81*2|^3 = 959*359096+704 by Lm14,Lm3;
        then 2|^(14*4+3),704 are_congruent_mod 959 &
          704*704*704*704*2|^3 = 959*2049094635+683 by Lem1;
        then 2|^(59*4+3),683 are_congruent_mod 959 &
          683*683*683*683*2|^3 = 959*1815324188+676 by LemS;
        then 2|^(239*4+3),676 are_congruent_mod 959 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 960;
        2|^15 = 960*34+128 &
          128*128*128*128*2|^0 = 960*279620+256 by Lm15,Lm1;
        then 2|^(15*4+0),256 are_congruent_mod 960 &
          256*256*256*256*2|^0 = 960*4473924+256 by Lem1;
        then 2|^(60*4+0),256 are_congruent_mod 960 &
          256*256*256*256*2|^0 = 960*4473924+256 by LemS;
        then 2|^(240*4+0),256 are_congruent_mod 960 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 961;
        2|^15 = 961*34+94 & 94*94*94*94*2|^0 = 961*81243+373 by Lm15,Lm1;
        then 2|^(15*4+0),373 are_congruent_mod 961 &
          373*373*373*373*2|^0 = 961*20142433+528 by Lem1;
        then 2|^(60*4+0),528 are_congruent_mod 961 &
          528*528*528*528*2|^1 = 961*161749258+374 by LemS;
        then 2|^(240*4+1),374 are_congruent_mod 961 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 962;
        2|^15 = 962*34+60 & 60*60*60*60*2|^0 = 962*13471+898 by Lm15,Lm1;
        then 2|^(15*4+0),898 are_congruent_mod 962 &
          898*898*898*898*2|^0 = 962*675974439+898 by Lem1;
        then 2|^(60*4+0),898 are_congruent_mod 962 &
          898*898*898*898*2|^2 = 962*(11199*241441)+706 by LemS,Lm2;
        then 2|^(240*4+2),706 are_congruent_mod 962 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 963;
        2|^15 = 963*34+26 & 26*26*26*26*2|^0 = 963*474+514 by Lm15,Lm1;
        then 2|^(15*4+0),514 are_congruent_mod 963 &
          514*514*514*514*2|^0 = 963*72481335+811 by Lem1;
        then 2|^(60*4+0),811 are_congruent_mod 963 &
          811*811*811*811*2|^3 = 963*(46444*77378)+512 by LemS,Lm3;
        then 2|^(240*4+3),512 are_congruent_mod 963 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 964;
        2|^15 = 964*33+956 &
          956*956*956*956*2|^0 = 964*866472004+240 by Lm15,Lm1;
        then 2|^(15*4+0),240 are_congruent_mod 964 &
          240*240*240*240*2|^1 = 964*6883319+484 by Lem1;
        then 2|^(60*4+1),484 are_congruent_mod 964 &
          484*484*484*484*2|^0 = 964*56925180+16 by LemS,Lm1;
        then 2|^(241*4+0),16 are_congruent_mod 964 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 965;
        2|^15 = 965*33+923 &
          923*923*923*923*2|^0 = 965*752106757+536 by Lm15,Lm1;
        then 2|^(15*4+0),536 are_congruent_mod 965 &
          536*536*536*536*2|^1 = 965*171065267+577 by Lem1;
        then 2|^(60*4+1),577 are_congruent_mod 965 &
          577*577*577*577*2|^1 = 965*229723770+32 by LemS;
        then 2|^(241*4+1),32 are_congruent_mod 965 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 966;
        2|^15 = 966*33+890 &
          890*890*890*890*2|^0 = 966*649505600+400 by Lm15,Lm1;
        then 2|^(15*4+0),400 are_congruent_mod 966 &
          400*400*400*400*2|^1 = 966*53002070+380 by Lem1;
        then 2|^(60*4+1),380 are_congruent_mod 966 &
          380*380*380*380*2|^2 = 966*86341035+190 by LemS,Lm2;
        then 2|^(241*4+2),190 are_congruent_mod 966 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 967;
        hence thesis by XPRIMES1:967;
      end;
      suppose A: n = 968;
        2|^15 = 968*33+824 &
          824*824*824*824*2|^0 = 968*476248355+936 by Lm15,Lm1;
        then 2|^(15*4+0),936 are_congruent_mod 968 &
          936*936*936*936*2|^2 = 968*(284*11167853)+928 by Lem1,Lm2;
        then 2|^(60*4+2),928 are_congruent_mod 968 &
          928*928*928*928*2|^0 = 968*766154836+608 by LemS,Lm1;
        then 2|^(242*4+0),608 are_congruent_mod 968 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 969;
        2|^15 = 969*33+791 &
          791*791*791*791*2|^0 = 969*404000736+577 by Lm15,Lm1;
        then 2|^(15*4+0),577 are_congruent_mod 969 &
          577*577*577*577*2|^2 = 969*457550955+769 by Lem1,Lm2;
        then 2|^(60*4+2),769 are_congruent_mod 969 &
          769*769*769*769*2|^1 = 969*721791191+563 by LemS;
        then 2|^(242*4+1),563 are_congruent_mod 969 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 970;
        2|^15 = 970*33+758 &
          758*758*758*758*2|^0 = 970*340333804+216 by Lm15,Lm1;
        then 2|^(15*4+0),216 are_congruent_mod 970 &
          216*216*216*216*2|^2 = 970*8976422+4 by Lem1,Lm2;
        then 2|^(60*4+2),4 are_congruent_mod 970 &
          4*4*4*4*2|^2 = 970*1+54 by LemS;
        then 2|^(242*4+2),54 are_congruent_mod 970 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 971;
        hence thesis by XPRIMES1:971;
      end;
      suppose A: n = 972;
        2|^15 = 972*33+692 &
          692*692*692*692*2|^0 = 972*235916389+388 by Lm15,Lm1;
        then 2|^(15*4+0),388 are_congruent_mod 972 &
          388*388*388*388*2|^3 = 972*186530830+728 by Lem1,Lm3;
        then 2|^(60*4+3),728 are_congruent_mod 972 &
          728*728*728*728*2|^0 = 972*288974321+244 by LemS,Lm1;
        then 2|^(243*4+0),244 are_congruent_mod 972 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 973;
        2|^15 = 973*33+659 &
          659*659*659*659*2|^0 = 973*193833491+218 by Lm15,Lm1;
        then 2|^(15*4+0),218 are_congruent_mod 973 &
          218*218*218*218*2|^3 = 973*18569624+456 by Lem1,Lm3;
        then 2|^(60*4+3),456 are_congruent_mod 973 &
          456*456*456*456*2|^1 = 973*88874368+128 by LemS;
        then 2|^(243*4+1),128 are_congruent_mod 973 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 974;
        2|^15 = 974*33+626 &
          626*626*626*626*2|^0 = 974*157666118+444 by Lm15,Lm1;
        then 2|^(15*4+0),444 are_congruent_mod 974 &
          444*444*444*444*2|^3 = 974*319200020+488 by Lem1,Lm3;
        then 2|^(60*4+3),488 are_congruent_mod 974 &
          488*488*488*488*2|^2 = 974*232905810+4 by LemS,Lm2;
        then 2|^(243*4+2),4 are_congruent_mod 974 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 975;
        2|^15 = 975*33+593 &
          593*593*593*593*2|^0 = 975*126827712+1 by Lm15,Lm1;
        then 2|^(15*4+0),1 are_congruent_mod 975 &
          1*1*1*1*2|^3 = 975*0+8 by Lem1,Lm3;
        then 2|^(60*4+3),8 are_congruent_mod 975 &
          8*8*8*8*2|^3 = 975*33+593 by LemS;
        then 2|^(243*4+3),593 are_congruent_mod 975 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 976;
        2|^15 = 976*33+560 &
          560*560*560*560*2|^1 = 976*201526557+368 by Lm15;
        then 2|^(15*4+1),368 are_congruent_mod 976 &
          368*368*368*368*2|^0 = 976*18790635+16 by Lem1,Lm1;
        then 2|^(61*4+0),16 are_congruent_mod 976 &
          16*16*16*16*2|^0 = 976*67+144 by LemS;
        then 2|^(244*4+0),144 are_congruent_mod 976 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 977;
        hence thesis by XPRIMES1:977;
      end;
      suppose A: n = 978;
        2|^15 = 978*33+494 &
          494*494*494*494*2|^1 = 978*121786440+272 by Lm15;
        then 2|^(15*4+1),272 are_congruent_mod 978 &
          272*272*272*272*2|^0 = 978*5596760+976 by Lem1,Lm1;
        then 2|^(61*4+0),976 are_congruent_mod 978 &
          976*976*976*976*2|^2 = 978*(47629*77920)+64 by LemS,Lm2;
        then 2|^(244*4+2),64 are_congruent_mod 978 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 979;
        2|^15 = 979*33+461 &
          461*461*461*461*2|^1 = 979*92267978+420 by Lm15;
        then 2|^(15*4+1),420 are_congruent_mod 979 &
          420*420*420*420*2|^0 = 979*31784433+93 by Lem1,Lm1;
        then 2|^(61*4+0),93 are_congruent_mod 979 &
          93*93*93*93*2|^3 = 979*611278+446 by LemS,Lm3;
        then 2|^(244*4+3),446 are_congruent_mod 979 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 980;
        2|^15 = 980*33+428 &
          428*428*428*428*2|^1 = 980*68482403+772 by Lm15;
        then 2|^(15*4+1),772 are_congruent_mod 980 &
          772*772*772*772*2|^1 = 980*724891690+312 by Lem1;
        then 2|^(61*4+1),312 are_congruent_mod 980 &
          312*312*312*312*2|^0 = 980*9669239+116 by LemS,Lm1;
        then 2|^(245*4+0),116 are_congruent_mod 980 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 981;
        2|^15 = 981*33+395 &
          395*395*395*395*2|^1 = 981*49630582+308 by Lm15;
        then 2|^(15*4+1),308 are_congruent_mod 981 &
          308*308*308*308*2|^1 = 981*18346949+23 by Lem1;
        then 2|^(61*4+1),23 are_congruent_mod 981 &
          23*23*23*23*2|^1 = 981*570+512 by LemS;
        then 2|^(245*4+1),512 are_congruent_mod 981 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 982;
        2|^15 = 982*33+362 &
          362*362*362*362*2|^1 = 982*34974602+708 by Lm15;
        then 2|^(15*4+1),708 are_congruent_mod 982 &
          708*708*708*708*2|^1 = 982*511742561+490 by Lem1;
        then 2|^(61*4+1),490 are_congruent_mod 982 &
          490*490*490*490*2|^2 = 982*234818778+4 by LemS,Lm2;
        then 2|^(245*4+2),4 are_congruent_mod 982 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 983;
        hence thesis by XPRIMES1:983;
      end;
      suppose A: n = 984;
        2|^15 = 984*33+296 &
          296*296*296*296*2|^1 = 984*15602771+248 by Lm15;
        then 2|^(15*4+1),248 are_congruent_mod 984 &
          248*248*248*248*2|^2 = 984*15377000+64 by Lem1,Lm2;
        then 2|^(61*4+2),64 are_congruent_mod 984 &
          64*64*64*64*2|^0 = 984*17050+16 by LemS,Lm1;
        then 2|^(246*4+0),16 are_congruent_mod 984 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 985;
        2|^15 = 985*33+263 &
          263*263*263*263*2|^1 = 985*9714417+377 by Lm15;
        then 2|^(15*4+1),377 are_congruent_mod 985 &
          377*377*377*377*2|^2 = 985*82033107+169 by Lem1,Lm2;
        then 2|^(61*4+2),169 are_congruent_mod 985 &
          169*169*169*169*2|^1 = 985*1656306+32 by LemS;
        then 2|^(246*4+1),32 are_congruent_mod 985 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 986;
        2|^15 = 986*33+230 &
          230*230*230*230*2|^1 = 986*5676288+32 by Lm15;
        then 2|^(15*4+1),32 are_congruent_mod 986 &
          32*32*32*32*2|^2 = 986*4253+846 by Lem1,Lm2;
        then 2|^(61*4+2),846 are_congruent_mod 986 &
          846*846*846*846*2|^2 = 986*2078090842+412 by LemS;
        then 2|^(246*4+2),412 are_congruent_mod 986 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 987;
        2|^15 = 987*33+197 &
          197*197*197*197*2|^1 = 987*3051952+338 by Lm15;
        then 2|^(15*4+1),338 are_congruent_mod 987 &
          338*338*338*338*2|^2 = 987*52894393+253 by Lem1,Lm2;
        then 2|^(61*4+2),253 are_congruent_mod 987 &
          253*253*253*253*2|^3 = 987*33208932+764 by LemS,Lm3;
        then 2|^(246*4+3),764 are_congruent_mod 987 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 988;
        2|^15 = 988*33+164 &
          164*164*164*164*2|^1 = 988*1464361+964 by Lm15;
        then 2|^(15*4+1),964 are_congruent_mod 988 &
          964*964*964*964*2|^3 = 988*(19638*356077)+440 by Lem1,Lm3;
        then 2|^(61*4+3),440 are_congruent_mod 988 &
          440*440*440*440*2|^0 = 988*37936194+328 by LemS,Lm1;
        then 2|^(247*4+0),328 are_congruent_mod 988 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 989;
        2|^15 = 989*33+131 &
          131*131*131*131*2|^1 = 989*595550+892 by Lm15;
        then 2|^(15*4+1),892 are_congruent_mod 989 &
          892*892*892*892*2|^3 = 989*(10663*480257)+469 by Lem1,Lm3;
        then 2|^(61*4+3),469 are_congruent_mod 989 &
          469*469*469*469*2|^1 = 989*97841944+426 by LemS;
        then 2|^(247*4+1),426 are_congruent_mod 989 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 990;
        2|^15 = 990*33+98 & 98*98*98*98*2|^1 = 990*186337+2 by Lm15;
        then 2|^(15*4+1),2 are_congruent_mod 990 &
          2*2*2*2*2|^3 = 990*0+128 by Lem1,Lm3;
        then 2|^(61*4+3),128 are_congruent_mod 990 &
          128*128*128*128*2|^2 = 990*1084587+694 by LemS,Lm2;
        then 2|^(247*4+2),694 are_congruent_mod 990 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 991;
        hence thesis by XPRIMES1:991;
      end;
      suppose A: n = 992;
        2|^15 = 992*33+32 & 32*32*32*32*2|^2 = 992*4228+128 by Lm15,Lm2;
        then 2|^(15*4+2),128 are_congruent_mod 992 &
          128*128*128*128*2|^0 = 992*270600+256 by Lem1,Lm1;
        then 2|^(62*4+0),256 are_congruent_mod 992 &
          256*256*256*256*2|^0 = 992*4329604+128 by LemS;
        then 2|^(248*4+0),128 are_congruent_mod 992 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 993;
        2|^15 = 993*32+992 &
          992*992*992*992*2|^2 = 993*(40580*96127)+4 by Lm15,Lm2;
        then 2|^(15*4+2),4 are_congruent_mod 993 &
          4*4*4*4*2|^0 = 993*0+256 by Lem1,Lm1;
        then 2|^(62*4+0),256 are_congruent_mod 993 &
          256*256*256*256*2|^1 = 993*8650488+8 by LemS;
        then 2|^(248*4+1),8 are_congruent_mod 993 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 994;
        2|^15 = 994*32+960 &
          960*960*960*960*2|^2 = 994*(3*1139297867)+606 by Lm15,Lm2;
        then 2|^(15*4+2),606 are_congruent_mod 994 &
          606*606*606*606*2|^0 = 994*135676337+718 by Lem1,Lm1;
        then 2|^(62*4+0),718 are_congruent_mod 994 &
          718*718*718*718*2|^2 = 994*1069476839+338 by LemS,Lm2;
        then 2|^(248*4+2),338 are_congruent_mod 994 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 995;
        2|^15 = 995*32+928 &
          928*928*928*928*2|^2 = 995*(53661*55561)+529 by Lm15,Lm2;
        then 2|^(15*4+2),529 are_congruent_mod 995 &
          529*529*529*529*2|^0 = 995*78704507+816 by Lem1,Lm1;
        then 2|^(62*4+0),816 are_congruent_mod 995 &
          816*816*816*816*2|^3 = 995*(50156*71073)+828 by LemS,Lm3;
        then 2|^(248*4+3),828 are_congruent_mod 995 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 996;
        2|^15 = 996*32+896 &
          896*896*896*896*2|^2 = 996*(41050*63055)+424 by Lm15,Lm2;
        then 2|^(15*4+2),424 are_congruent_mod 996 &
          424*424*424*424*2|^1 = 996*64898414+8 by Lem1;
        then 2|^(62*4+1),8 are_congruent_mod 996 &
          8*8*8*8*2|^0 = 996*4+112 by LemS,Lm1;
        then 2|^(249*4+0),112 are_congruent_mod 996 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 997;
        hence thesis by XPRIMES1:997;
      end;
      suppose A: n = 998;
        2|^15 = 998*32+832 &
          832*832*832*832*2|^2 = 998*1920537339+382 by Lm15,Lm2;
        then 2|^(15*4+2),382 are_congruent_mod 998 &
          382*382*382*382*2|^1 = 998*42672973+498 by Lem1;
        then 2|^(62*4+1),498 are_congruent_mod 998 &
          498*498*498*498*2|^2 = 998*246516970+4 by LemS,Lm2;
        then 2|^(249*4+2),4 are_congruent_mod 998 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 999;
        2|^15 = 999*32+800 &
          800*800*800*800*2|^2 = 999*1640040040+40 by Lm15,Lm2;
        then 2|^(15*4+2),40 are_congruent_mod 999 &
          40*40*40*40*2|^1 = 999*5125+125 by Lem1;
        then 2|^(62*4+1),125 are_congruent_mod 999 &
          125*125*125*125*2|^3 = 999*1955080+80 by LemS,Lm3;
        then 2|^(249*4+3),80 are_congruent_mod 999 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1000;
        2|^15 = 1000*32+768 &
          768*768*768*768*2|^2 = 1000*1391569403+904 by Lm15,Lm2;
        then 2|^(15*4+2),904 are_congruent_mod 1000 &
          904*904*904*904*2|^2 = 1000*(2*1335683981)+624 by Lem1;
        then 2|^(62*4+2),624 are_congruent_mod 1000 &
          624*624*624*624*2|^0 = 1000*151613669+376 by LemS,Lm1;
        then 2|^(250*4+0),376 are_congruent_mod 1000 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1001;
        2|^15 = 1001*32+736 &
          736*736*736*736*2|^2 = 1001*1172565660+4 by Lm15,Lm2;
        then 2|^(15*4+2),4 are_congruent_mod 1001 &
          4*4*4*4*2|^2 = 1001*1+23 by Lem1;
        then 2|^(62*4+2),23 are_congruent_mod 1001 &
          23*23*23*23*2|^1 = 1001*559+123 by LemS;
        then 2|^(250*4+1),123 are_congruent_mod 1001 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1002;
        2|^15 = 1002*32+704 &
          704*704*704*704*2|^2 = 1002*980579718+388 by Lm15,Lm2;
        then 2|^(15*4+2),388 are_congruent_mod 1002 &
          388*388*388*388*2|^2 = 1002*90473037+670 by Lem1;
        then 2|^(62*4+2),670 are_congruent_mod 1002 &
          670*670*670*670*2|^2 = 1002*804435968+64 by LemS;
        then 2|^(250*4+2),64 are_congruent_mod 1002 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1003;
        2|^15 = 1003*32+672 &
          672*672*672*672*2|^2 = 1003*813272618+370 by Lm15,Lm2;
        then 2|^(15*4+2),370 are_congruent_mod 1003 &
          370*370*370*370*2|^2 = 1003*74742213+361 by Lem1;
        then 2|^(62*4+2),361 are_congruent_mod 1003 &
          361*361*361*361*2|^3 = 1003*135462117+977 by LemS,Lm3;
        then 2|^(250*4+3),977 are_congruent_mod 1003 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1004;
        2|^15 = 1004*32+640 &
          640*640*640*640*2|^2 = 1004*668414980+80 by Lm15,Lm2;
        then 2|^(15*4+2),80 are_congruent_mod 1004 &
          80*80*80*80*2|^3 = 1004*326374+504 by Lem1,Lm3;
        then 2|^(62*4+3),504 are_congruent_mod 1004 &
          504*504*504*504*2|^0 = 1004*64267060+16 by LemS,Lm1;
        then 2|^(251*4+0),16 are_congruent_mod 1004 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 1004 < n & n <= 1104;
      then 1004+1 <= n & n <= 1104 by NAT_1:13;
      then n = 1005 or ... or n = 1104;
      then per cases;
      suppose A: n = 1005;
        2|^15 = 1005*32+608 &
          608*608*608*608*2|^2 = 1005*543886459+289 by Lm15,Lm2;
        then 2|^(15*4+2),289 are_congruent_mod 1005 &
          289*289*289*289*2|^3 = 1005*55528417+443 by Lem1,Lm3;
        then 2|^(62*4+3),443 are_congruent_mod 1005 &
          443*443*443*443*2|^1 = 1005*76644119+407 by LemS;
        then 2|^(251*4+1),407 are_congruent_mod 1005 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1006;
        2|^15 = 1006*32+576 &
          576*576*576*576*2|^2 = 1006*437675205+474 by Lm15,Lm2;
        then 2|^(15*4+2),474 are_congruent_mod 1006 &
          474*474*474*474*2|^3 = 1006*401425884+504 by Lem1,Lm3;
        then 2|^(62*4+3),504 are_congruent_mod 1006 &
          504*504*504*504*2|^2 = 1006*256557170+4 by LemS,Lm2;
        then 2|^(251*4+2),4 are_congruent_mod 1006 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1007;
        2|^15 = 1007*32+544 &
          544*544*544*544*2|^2 = 1007*347877323+123 by Lm15,Lm2;
        then 2|^(15*4+2),123 are_congruent_mod 1007 &
          123*123*123*123*2|^3 = 1007*1818364+580 by Lem1,Lm3;
        then 2|^(62*4+3),580 are_congruent_mod 1007 &
          580*580*580*580*2|^3 = 1007*899026494+542 by LemS;
        then 2|^(251*4+3),542 are_congruent_mod 1007 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1008;
        2|^15 = 1008*32+512 &
          512*512*512*512*2|^3 = 1008*545392672+512 by Lm15,Lm3;
        then 2|^(15*4+3),512 are_congruent_mod 1008 &
          512*512*512*512*2|^0 = 1008*68174084+64 by Lem1,Lm1;
        then 2|^(63*4+0),64 are_congruent_mod 1008 &
          64*64*64*64*2|^0 = 1008*16644+64 by LemS;
        then 2|^(252*4+0),64 are_congruent_mod 1008 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1009;
        hence thesis by XPRIMES1:1009;
      end;
      suppose A: n = 1010;
        2|^15 = 1010*32+448 &
          448*448*448*448*2|^3 = 1010*319066103+898 by Lm15,Lm3;
        then 2|^(15*4+3),898 are_congruent_mod 1010 &
          898*898*898*898*2|^0 = 1010*643848921+1006 by Lem1,Lm1;
        then 2|^(63*4+0),1006 are_congruent_mod 1010 &
          1006*1006*1006*1006*2|^2 = 1010*(21503*188639)+14 by LemS,Lm2;
        then 2|^(252*4+2),14 are_congruent_mod 1010 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1011;
        2|^15 = 1011*32+416 &
          416*416*416*416*2|^3 = 1011*236980250+338 by Lm15,Lm3;
        then 2|^(15*4+3),338 are_congruent_mod 1011 &
          338*338*338*338*2|^0 = 1011*12909685+1 by Lem1,Lm1;
        then 2|^(63*4+0),1 are_congruent_mod 1011 &
          1*1*1*1*2|^3 = 1011*0+8 by LemS,Lm3;
        then 2|^(252*4+3),8 are_congruent_mod 1011 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1012;
        2|^15 = 1012*32+384 &
          384*384*384*384*2|^3 = 1012*171883572+624 by Lm15,Lm3;
        then 2|^(15*4+3),624 are_congruent_mod 1012 &
          624*624*624*624*2|^1 = 1012*299631757+668 by Lem1;
        then 2|^(63*4+1),668 are_congruent_mod 1012 &
          668*668*668*668*2|^0 = 1012*196754800+576 by LemS,Lm1;
        then 2|^(253*4+0),576 are_congruent_mod 1012 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1013;
        hence thesis by XPRIMES1:1013;
      end;
      suppose A: n = 1014;
        2|^15 = 1014*32+320 &
          320*320*320*320*2|^3 = 1014*82727889+554 by Lm15,Lm3;
        then 2|^(15*4+3),554 are_congruent_mod 1014 &
          554*554*554*554*2|^1 = 1014*185793749+626 by Lem1;
        then 2|^(63*4+1),626 are_congruent_mod 1014 &
          626*626*626*626*2|^2 = 1014*605786190+844 by LemS,Lm2;
        then 2|^(253*4+2),844 are_congruent_mod 1014 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1015;
        2|^15 = 1015*32+288 &
          288*288*288*288*2|^3 = 1015*54224292+708 by Lm15,Lm3;
        then 2|^(15*4+3),708 are_congruent_mod 1015 &
          708*708*708*708*2|^1 = 1015*495104626+2 by Lem1;
        then 2|^(63*4+1),2 are_congruent_mod 1015 &
          2*2*2*2*2|^3 = 1015*0+128 by LemS,Lm3;
        then 2|^(253*4+3),128 are_congruent_mod 1015 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1016;
        2|^15 = 1016*32+256 &
          256*256*256*256*2|^3 = 1016*33818640+128 by Lm15,Lm3;
        then 2|^(15*4+3),128 are_congruent_mod 1016 &
          128*128*128*128*2|^2 = 1016*1056832+512 by Lem1,Lm2;
        then 2|^(63*4+2),512 are_congruent_mod 1016 &
          512*512*512*512*2|^0 = 1016*67637280+256 by LemS,Lm1;
        then 2|^(254*4+0),256 are_congruent_mod 1016 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1017;
        2|^15 = 1017*32+224 &
          224*224*224*224*2|^3 = 1017*19804373+467 by Lm15,Lm3;
        then 2|^(15*4+3),467 are_congruent_mod 1017 &
          467*467*467*467*2|^2 = 1017*187071040+4 by Lem1,Lm2;
        then 2|^(63*4+2),4 are_congruent_mod 1017 &
          4*4*4*4*2|^1 = 1017*0+512 by LemS;
        then 2|^(254*4+1),512 are_congruent_mod 1017 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1018;
        2|^15 = 1018*32+192 &
          192*192*192*192*2|^3 = 1018*10679406+660 by Lm15,Lm3;
        then 2|^(15*4+3),660 are_congruent_mod 1018 &
          660*660*660*660*2|^2 = 1018*745569194+508 by Lem1,Lm2;
        then 2|^(63*4+2),508 are_congruent_mod 1018 &
          508*508*508*508*2|^2 = 1018*261677910+4 by LemS;
        then 2|^(254*4+2),4 are_congruent_mod 1018 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1019;
        hence thesis by XPRIMES1:1019;
      end;
      suppose A: n = 1020;
        2|^15 = 1020*32+128 &
          128*128*128*128*2|^3 = 1020*2105376+128 by Lm15,Lm3;
        then 2|^(15*4+3),128 are_congruent_mod 1020 &
          128*128*128*128*2|^3 = 1020*2105376+128 by Lem1;
        then 2|^(63*4+3),128 are_congruent_mod 1020 &
          128*128*128*128*2|^0 = 1020*263172+16 by LemS,Lm1;
        then 2|^(255*4+0),16 are_congruent_mod 1020 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1021;
        hence thesis by XPRIMES1:1021;
      end;
      suppose A: n = 1022;
        2|^15 = 1022*32+64 & 64*64*64*64*2|^3 = 1022*131328+512 by Lm15,Lm3;
        then 2|^(15*4+3),512 are_congruent_mod 1022 &
          512*512*512*512*2|^3 = 1022*537921540+8 by Lem1;
        then 2|^(63*4+3),8 are_congruent_mod 1022 &
          8*8*8*8*2|^2 = 1022*16+32 by LemS,Lm2;
        then 2|^(255*4+2),32 are_congruent_mod 1022 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1023;
        2|^15 = 1023*32+32 & 32*32*32*32*2|^3 = 1023*8200+8 by Lm15,Lm3;
        then 2|^(15*4+3),8 are_congruent_mod 1023 &
          8*8*8*8*2|^3 = 1023*32+32 by Lem1;
        then 2|^(63*4+3),32 are_congruent_mod 1023 &
          32*32*32*32*2|^3 = 1023*8200+8 by LemS;
        then 2|^(255*4+3),8 are_congruent_mod 1023 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1024;
        2|^16 = 1024*64+0 & 0*0*0*0*2|^0 = 1024*0+0 by Lm16;
        then 2|^(16*4+0),0 are_congruent_mod 1024 &
          0*0*0*0*2|^0 = 1024*0+0 by Lem1;
        then 2|^(64*4+0),0 are_congruent_mod 1024 &
          0*0*0*0*2|^0 = 1024*0+0 by LemS;
        then 2|^(256*4+0),0 are_congruent_mod 1024 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1025;
        2|^16 = 1025*63+961 &
          961*961*961*961*2|^0 = 1025*832088817+16 by Lm16,Lm1;
        then 2|^(16*4+0),16 are_congruent_mod 1025 &
          16*16*16*16*2|^0 = 1025*63+961 by Lem1;
        then 2|^(64*4+0),961 are_congruent_mod 1025 &
          961*961*961*961*2|^1 = 1025*1664177634+32 by LemS;
        then 2|^(256*4+1),32 are_congruent_mod 1025 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1026;
        2|^16 = 1026*63+898 &
          898*898*898*898*2|^0 = 1026*633808392+1024 by Lm16,Lm1;
        then 2|^(16*4+0),1024 are_congruent_mod 1026 &
          1024*1024*1024*1024*2|^0 = 1026*1071648760+16 by Lem1;
        then 2|^(64*4+0),16 are_congruent_mod 1026 &
          16*16*16*16*2|^2 = 1026*255+514 by LemS,Lm2;
        then 2|^(256*4+2),514 are_congruent_mod 1026 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1027;
        2|^16 = 1027*63+835 &
          835*835*835*835*2|^0 = 1027*473342454+367 by Lm16,Lm1;
        then 2|^(16*4+0),367 are_congruent_mod 1027 &
          367*367*367*367*2|^0 = 1027*17664193+510 by Lem1;
        then 2|^(64*4+0),510 are_congruent_mod 1027 &
          510*510*510*510*2|^3 = 1027*526987419+687 by LemS,Lm3;
        then 2|^(256*4+3),687 are_congruent_mod 1027 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1028;
        2|^16 = 1028*63+772 &
          772*772*772*772*2|^0 = 1028*345522303+772 by Lm16,Lm1;
        then 2|^(16*4+0),772 are_congruent_mod 1028 &
          772*772*772*772*2|^1 = 1028*691044607+516 by Lem1;
        then 2|^(64*4+1),516 are_congruent_mod 1028 &
          516*516*516*516*2|^0 = 1028*68961340+16 by LemS,Lm1;
        then 2|^(257*4+0),16 are_congruent_mod 1028 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1029;
        2|^16 = 1029*63+709 &
          709*709*709*709*2|^0 = 1029*245566751+982 by Lm16,Lm1;
        then 2|^(16*4+0),982 are_congruent_mod 1029 &
          982*982*982*982*2|^1 = 1029*1807426194+326 by Lem1;
        then 2|^(64*4+1),326 are_congruent_mod 1029 &
          326*326*326*326*2|^1 = 1029*21952552+344 by LemS;
        then 2|^(257*4+1),344 are_congruent_mod 1029 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1030;
        2|^16 = 1030*63+646 &
          646*646*646*646*2|^0 = 1030*169080236+776 by Lm16,Lm1;
        then 2|^(16*4+0),776 are_congruent_mod 1030 &
          776*776*776*776*2|^1 = 1030*704108611+622 by Lem1;
        then 2|^(64*4+1),622 are_congruent_mod 1030 &
          622*622*622*622*2|^2 = 1030*581278560+1024 by LemS,Lm2;
        then 2|^(257*4+2),1024 are_congruent_mod 1030 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1031;
        hence thesis by XPRIMES1:1031;
      end;
      suppose A: n = 1032;
        2|^16 = 1032*63+520 &
          520*520*520*520*2|^0 = 1032*70848992+256 by Lm16,Lm1;
        then 2|^(16*4+0),256 are_congruent_mod 1032 &
          256*256*256*256*2|^2 = 1032*16647160+64 by Lem1,Lm2;
        then 2|^(64*4+2),64 are_congruent_mod 1032 &
          64*64*64*64*2|^0 = 1032*16256+1024 by LemS,Lm1;
        then 2|^(258*4+0),1024 are_congruent_mod 1032 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1033;
        hence thesis by XPRIMES1:1033;
      end;
      suppose A: n = 1034;
        2|^16 = 1034*63+394 &
          394*394*394*394*2|^0 = 1034*23305817+918 by Lm16,Lm1;
        then 2|^(16*4+0),918 are_congruent_mod 1034 &
          918*918*918*918*2|^2 = 1034*(42380*64826)+784 by Lem1,Lm2;
        then 2|^(64*4+2),784 are_congruent_mod 1034 &
          784*784*784*784*2|^2 = 1034*1461516434+588 by LemS;
        then 2|^(258*4+2),588 are_congruent_mod 1034 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1035;
        2|^16 = 1035*63+331 &
          331*331*331*331*2|^0 = 1035*11597693+466 by Lm16,Lm1;
        then 2|^(16*4+0),466 are_congruent_mod 1035 &
          466*466*466*466*2|^2 = 1035*182248225+469 by Lem1,Lm2;
        then 2|^(64*4+2),469 are_congruent_mod 1035 &
          469*469*469*469*2|^3 = 1035*373973654+278 by LemS,Lm3;
        then 2|^(258*4+3),278 are_congruent_mod 1035 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1036;
        2|^16 = 1036*63+268 &
          268*268*268*268*2|^0 = 1036*4979427+604 by Lm16,Lm1;
        then 2|^(16*4+0),604 are_congruent_mod 1036 &
          604*604*604*604*2|^3 = 1036*1027727520+128 by Lem1,Lm3;
        then 2|^(64*4+3),128 are_congruent_mod 1036 &
          128*128*128*128*2|^0 = 1036*259107+604 by LemS,Lm1;
        then 2|^(259*4+0),604 are_congruent_mod 1036 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1037;
        2|^16 = 1037*63+205 &
          205*205*205*205*2|^0 = 1037*1703086+443 by Lm16,Lm1;
        then 2|^(16*4+0),443 are_congruent_mod 1037 &
          443*443*443*443*2|^3 = 1037*297116065+603 by Lem1,Lm3;
        then 2|^(64*4+3),603 are_congruent_mod 1037 &
          603*603*603*603*2|^1 = 1037*254988437+593 by LemS;
        then 2|^(259*4+1),593 are_congruent_mod 1037 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1038;
        2|^16 = 1038*63+142 &
          142*142*142*142*2|^0 = 1038*391702+220 by Lm16,Lm1;
        then 2|^(16*4+0),220 are_congruent_mod 1038 &
          220*220*220*220*2|^3 = 1038*18054412+344 by Lem1,Lm3;
        then 2|^(64*4+3),344 are_congruent_mod 1038 &
          344*344*344*344*2|^2 = 1038*53963040+64 by LemS,Lm2;
        then 2|^(259*4+2),64 are_congruent_mod 1038 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1039;
        hence thesis by XPRIMES1:1039;
      end;
      suppose A: n = 1040;
        2|^16 = 1040*63+16 & 16*16*16*16*2|^1 = 1040*126+32 by Lm16;
        then 2|^(16*4+1),32 are_congruent_mod 1040 &
          32*32*32*32*2|^0 = 1040*1008+256 by Lem1,Lm1;
        then 2|^(65*4+0),256 are_congruent_mod 1040 &
          256*256*256*256*2|^0 = 1040*4129776+256 by LemS;
        then 2|^(260*4+0),256 are_congruent_mod 1040 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1041;
        2|^16 = 1041*62+994 &
          994*994*994*994*2|^1 = 1041*1875533404+1028 by Lm16;
        then 2|^(16*4+1),1028 are_congruent_mod 1041 &
          1028*1028*1028*1028*2|^0 = 1041*1072807322+454 by Lem1,Lm1;
        then 2|^(65*4+0),454 are_congruent_mod 1041 &
          454*454*454*454*2|^1 = 1041*81621144+8 by LemS;
        then 2|^(260*4+1),8 are_congruent_mod 1041 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1042;
        2|^16 = 1042*62+932 &
          932*932*932*932*2|^1 = 1042*1448191273+286 by Lm16;
        then 2|^(16*4+1),286 are_congruent_mod 1042 &
          286*286*286*286*2|^0 = 1042*6420907+522 by Lem1,Lm1;
        then 2|^(65*4+0),522 are_congruent_mod 1042 &
          522*522*522*522*2|^2 = 1042*285019310+4 by LemS,Lm2;
        then 2|^(260*4+2),4 are_congruent_mod 1042 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1043;
        2|^16 = 1043*62+870 &
          870*870*870*870*2|^1 = 1043*1098557257+949 by Lm16;
        then 2|^(16*4+1),949 are_congruent_mod 1043 &
          949*949*949*949*2|^0 = 1043*777643491+88 by Lem1,Lm1;
        then 2|^(65*4+0),88 are_congruent_mod 1043 &
          88*88*88*88*2|^3 = 1043*459977+277 by LemS,Lm3;
        then 2|^(260*4+3),277 are_congruent_mod 1043 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1044;
        2|^16 = 1044*62+808 &
          808*808*808*808*2|^1 = 1044*816535253+860 by Lm16;
        then 2|^(16*4+1),860 are_congruent_mod 1044 &
          860*860*860*860*2|^1 = 1044*1047908352+512 by Lem1;
        then 2|^(65*4+1),512 are_congruent_mod 1044 &
          512*512*512*512*2|^0 = 1044*65823253+604 by LemS,Lm1;
        then 2|^(261*4+0),604 are_congruent_mod 1044 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1045;
        2|^16 = 1045*62+746 &
          746*746*746*746*2|^1 = 1045*592746522+1022 by Lm16;
        then 2|^(16*4+1),1022 are_congruent_mod 1045 &
          1022*1022*1022*1022*2|^1 = 1045*2087936509+607 by Lem1;
        then 2|^(65*4+1),607 are_congruent_mod 1045 &
          607*607*607*607*2|^1 = 1045*259817541+857 by LemS;
        then 2|^(261*4+1),857 are_congruent_mod 1045 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1046;
        2|^16 = 1046*62+684 &
          684*684*684*684*2|^1 = 1046*418526265+282 by Lm16;
        then 2|^(16*4+1),282 are_congruent_mod 1046 &
          282*282*282*282*2|^1 = 1046*12091905+522 by Lem1;
        then 2|^(65*4+1),522 are_congruent_mod 1046 &
          522*522*522*522*2|^2 = 1046*283929370+4 by LemS,Lm2;
        then 2|^(261*4+2),4 are_congruent_mod 1046 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1047;
        2|^16 = 1047*62+622 &
          622*622*622*622*2|^1 = 1047*285920209+89 by Lm16;
        then 2|^(16*4+1),89 are_congruent_mod 1047 &
          89*89*89*89*2|^1 = 1047*119851+485 by Lem1;
        then 2|^(65*4+1),485 are_congruent_mod 1047 &
          485*485*485*485*2|^3 = 1047*422775936+8 by LemS,Lm3;
        then 2|^(261*4+3),8 are_congruent_mod 1047 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1048;
        2|^16 = 1048*62+560 &
          560*560*560*560*2|^1 = 1048*187681221+392 by Lm16;
        then 2|^(16*4+1),392 are_congruent_mod 1048 &
          392*392*392*392*2|^2 = 1048*90124522+528 by Lem1,Lm2;
        then 2|^(65*4+2),528 are_congruent_mod 1048 &
          528*528*528*528*2|^0 = 1048*74160800+256 by LemS,Lm1;
        then 2|^(262*4+0),256 are_congruent_mod 1048 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1049;
        hence thesis by XPRIMES1:1049;
      end;
      suppose A: n = 1050;
        2|^16 = 1050*62+436 &
          436*436*436*436*2|^1 = 1050*68831408+32 by Lm16;
        then 2|^(16*4+1),32 are_congruent_mod 1050 &
          32*32*32*32*2|^2 = 1050*3994+604 by Lem1,Lm2;
        then 2|^(65*4+2),604 are_congruent_mod 1050 &
          604*604*604*604*2|^2 = 1050*507012243+274 by LemS;
        then 2|^(262*4+2),274 are_congruent_mod 1050 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1051;
        hence thesis by XPRIMES1:1051;
      end;
      suppose A: n = 1052;
        2|^16 = 1052*62+312 &
          312*312*312*312*2|^1 = 1052*18014932+208 by Lm16;
        then 2|^(16*4+1),208 are_congruent_mod 1052 &
          208*208*208*208*2|^3 = 1052*14234020+528 by Lem1,Lm3;
        then 2|^(65*4+3),528 are_congruent_mod 1052 &
          528*528*528*528*2|^0 = 1052*73878820+16 by LemS,Lm1;
        then 2|^(263*4+0),16 are_congruent_mod 1052 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1053;
        2|^16 = 1053*62+250 &
          250*250*250*250*2|^1 = 1053*7419278+266 by Lm16;
        then 2|^(16*4+1),266 are_congruent_mod 1053 &
          266*266*266*266*2|^3 = 1053*38035415+293 by Lem1,Lm3;
        then 2|^(65*4+3),293 are_congruent_mod 1053 &
          293*293*293*293*2|^1 = 1053*13998197+161 by LemS;
        then 2|^(263*4+1),161 are_congruent_mod 1053 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1054;
        2|^16 = 1054*62+188 &
          188*188*188*188*2|^1 = 1054*2370395+342 by Lm16;
        then 2|^(16*4+1),342 are_congruent_mod 1054 &
          342*342*342*342*2|^3 = 1054*103837398+876 by Lem1,Lm3;
        then 2|^(65*4+3),876 are_congruent_mod 1054 &
          876*876*876*876*2|^2 = 1054*(5*446957059)+574 by LemS,Lm2;
        then 2|^(263*4+2),574 are_congruent_mod 1054 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1055;
        2|^16 = 1055*62+126 &
          126*126*126*126*2|^1 = 1055*477814+982 by Lm16;
        then 2|^(16*4+1),982 are_congruent_mod 1055 &
          982*982*982*982*2|^3 = 1055*(70817*99574)+118 by Lem1,Lm3;
        then 2|^(65*4+3),118 are_congruent_mod 1055 &
          118*118*118*118*2|^3 = 1055*1470163+243 by LemS;
        then 2|^(263*4+3),243 are_congruent_mod 1055 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1056;
        2|^16 = 1056*62+64 & 64*64*64*64*2|^2 = 1056*63550+64 by Lm16,Lm2;
        then 2|^(16*4+2),64 are_congruent_mod 1056 &
          64*64*64*64*2|^0 = 1056*15887+544 by Lem1,Lm1;
        then 2|^(66*4+0),544 are_congruent_mod 1056 &
          544*544*544*544*2|^0 = 1056*82933822+64 by LemS;
        then 2|^(264*4+0),64 are_congruent_mod 1056 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1057;
        2|^16 = 1057*62+2 & 2*2*2*2*2|^2 = 1057*0+64 by Lm16,Lm2;
        then 2|^(16*4+2),64 are_congruent_mod 1057 &
          64*64*64*64*2|^0 = 1057*15872+512 by Lem1,Lm1;
        then 2|^(66*4+0),512 are_congruent_mod 1057 &
          512*512*512*512*2|^1 = 1057*130027392+128 by LemS;
        then 2|^(264*4+1),128 are_congruent_mod 1057 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1058;
        2|^16 = 1058*61+998 &
          998*998*998*998*2|^2 = 1058*(18434*203459)+116 by Lm16,Lm2;
        then 2|^(16*4+2),116 are_congruent_mod 1058 &
          116*116*116*116*2|^0 = 1058*171137+990 by Lem1,Lm1;
        then 2|^(66*4+0),990 are_congruent_mod 1058 &
          990*990*990*990*2|^2 = 1058*(4*907935737)+1016 by LemS,Lm2;
        then 2|^(264*4+2),1016 are_congruent_mod 1058 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1059;
        2|^16 = 1059*61+937 &
          937*937*937*937*2|^2 = 1059*(8*363942193)+748 by Lm16,Lm2;
        then 2|^(16*4+2),748 are_congruent_mod 1059 &
          748*748*748*748*2|^0 = 1059*295604085+1 by Lem1,Lm1;
        then 2|^(66*4+0),1 are_congruent_mod 1059 &
          1*1*1*1*2|^3 = 1059*0+8 by LemS,Lm3;
        then 2|^(264*4+3),8 are_congruent_mod 1059 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1060;
        2|^16 = 1060*61+876 &
          876*876*876*876*2|^2 = 1060*(14479*153473)+484 by Lm16,Lm2;
        then 2|^(16*4+2),484 are_congruent_mod 1060 &
          484*484*484*484*2|^1 = 1060*103539384+32 by Lem1;
        then 2|^(66*4+1),32 are_congruent_mod 1060 &
          32*32*32*32*2|^0 = 1060*989+236 by LemS,Lm1;
        then 2|^(265*4+0),236 are_congruent_mod 1060 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1061;
        hence thesis by XPRIMES1:1061;
      end;
      suppose A: n = 1062;
        2|^16 = 1062*61+754 &
          754*754*754*754*2|^2 = 1062*1217365130+964 by Lm16,Lm2;
        then 2|^(16*4+2),964 are_congruent_mod 1062 &
          964*964*964*964*2|^1 = 1062*1626348503+1046 by Lem1;
        then 2|^(66*4+1),1046 are_congruent_mod 1062 &
          1046*1046*1046*1046*2|^2 = 1062*(82*54985523)+892 by LemS,Lm2;
        then 2|^(265*4+2),892 are_congruent_mod 1062 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1063;
        hence thesis by XPRIMES1:1063;
      end;
      suppose A: n = 1064;
        2|^16 = 1064*61+632 &
          632*632*632*632*2|^2 = 1064*599772675+904 by Lm16,Lm2;
        then 2|^(16*4+2),904 are_congruent_mod 1064 &
          904*904*904*904*2|^2 = 1064*(75*33475789)+424 by Lem1;
        then 2|^(66*4+2),424 are_congruent_mod 1064 &
          424*424*424*424*2|^0 = 1064*30375385+536 by LemS,Lm1;
        then 2|^(266*4+0),536 are_congruent_mod 1064 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1065;
        2|^16 = 1065*61+571 &
          571*571*571*571*2|^2 = 1065*399259093+679 by Lm16,Lm2;
        then 2|^(16*4+2),679 are_congruent_mod 1065 &
          679*679*679*679*2|^2 = 1065*798342924+664 by Lem1;
        then 2|^(66*4+2),664 are_congruent_mod 1065 &
          664*664*664*664*2|^1 = 1065*365050296+392 by LemS;
        then 2|^(266*4+1),392 are_congruent_mod 1065 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1066;
        2|^16 = 1066*61+510 &
          510*510*510*510*2|^2 = 1066*253853696+64 by Lm16,Lm2;
        then 2|^(16*4+2),64 are_congruent_mod 1066 &
          64*64*64*64*2|^2 = 1066*62953+966 by Lem1;
        then 2|^(66*4+2),966 are_congruent_mod 1066 &
          966*966*966*966*2|^2 = 1066*(2*1633733809)+556 by LemS;
        then 2|^(266*4+2),556 are_congruent_mod 1066 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1067;
        2|^16 = 1067*61+449 &
          449*449*449*449*2|^2 = 1067*152363498+438 by Lm16,Lm2;
        then 2|^(16*4+2),438 are_congruent_mod 1067 &
          438*438*438*438*2|^2 = 1067*137972334+966 by Lem1;
        then 2|^(66*4+2),966 are_congruent_mod 1067 &
          966*966*966*966*2|^3 = 1067*(383*17046503)+205 by LemS,Lm3;
        then 2|^(266*4+3),205 are_congruent_mod 1067 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1068;
        2|^16 = 1068*61+388 &
          388*388*388*388*2|^2 = 1068*84882007+268 by Lm16,Lm2;
        then 2|^(16*4+2),268 are_congruent_mod 1068 &
          268*268*268*268*2|^3 = 1068*38641850+8 by Lem1,Lm3;
        then 2|^(66*4+3),8 are_congruent_mod 1068 &
          8*8*8*8*2|^0 = 1068*3+892 by LemS,Lm1;
        then 2|^(267*4+0),892 are_congruent_mod 1068 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1069;
        hence thesis by XPRIMES1:1069;
      end;
      suppose A: n = 1070;
        2|^16 = 1070*61+266 &
          266*266*266*266*2|^2 = 1070*18715557+154 by Lm16,Lm2;
        then 2|^(16*4+2),154 are_congruent_mod 1070 &
          154*154*154*154*2|^3 = 1070*4205223+638 by Lem1,Lm3;
        then 2|^(66*4+3),638 are_congruent_mod 1070 &
          638*638*638*638*2|^2 = 1070*619382496+1024 by LemS,Lm2;
        then 2|^(267*4+2),1024 are_congruent_mod 1070 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1071;
        2|^16 = 1071*61+205 &
          205*205*205*205*2|^2 = 1071*6596080+820 by Lm16,Lm2;
        then 2|^(16*4+2),820 are_congruent_mod 1071 &
          820*820*820*820*2|^3 = 1071*(55588*60754)+8 by Lem1,Lm3;
        then 2|^(66*4+3),8 are_congruent_mod 1071 &
          8*8*8*8*2|^3 = 1071*30+638 by LemS;
        then 2|^(267*4+3),638 are_congruent_mod 1071 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1072;
        2|^16 = 1072*61+144 &
          144*144*144*144*2|^3 = 1072*3208818+672 by Lm16,Lm3;
        then 2|^(16*4+3),672 are_congruent_mod 1072 &
          672*672*672*672*2|^0 = 1072*190231445+16 by Lem1,Lm1;
        then 2|^(67*4+0),16 are_congruent_mod 1072 &
          16*16*16*16*2|^0 = 1072*61+144 by LemS;
        then 2|^(268*4+0),144 are_congruent_mod 1072 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1073;
        2|^16 = 1073*61+83 & 83*83*83*83*2|^3 = 1073*353836+540 by Lm16,Lm3;
        then 2|^(16*4+3),540 are_congruent_mod 1073 &
          540*540*540*540*2|^0 = 1073*79245629+83 by Lem1,Lm1;
        then 2|^(67*4+0),83 are_congruent_mod 1073 &
          83*83*83*83*2|^1 = 1073*88459+135 by LemS;
        then 2|^(268*4+1),135 are_congruent_mod 1073 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1074;
        2|^16 = 1074*61+22 & 22*22*22*22*2|^3 = 1074*1744+992 by Lm16,Lm3;
        then 2|^(16*4+3),992 are_congruent_mod 1074 &
          992*992*992*992*2|^0 = 1074*901659176+1072 by Lem1,Lm1;
        then 2|^(67*4+0),1072 are_congruent_mod 1074 &
          1072*1072*1072*1072*2|^2 = 1074*(17120*287297)+64 by LemS,Lm2;
        then 2|^(268*4+2),64 are_congruent_mod 1074 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1075;
        2|^16 = 1075*60+1036 &
          1036*1036*1036*1036*2|^3 = 1075*(10664*803897)+328 by Lm16,Lm3;
        then 2|^(16*4+3),328 are_congruent_mod 1075 &
          328*328*328*328*2|^0 = 1075*10766806+606 by Lem1,Lm1;
        then 2|^(67*4+0),606 are_congruent_mod 1075 &
          606*606*606*606*2|^3 = 1075*1003626267+543 by LemS,Lm3;
        then 2|^(268*4+3),543 are_congruent_mod 1075 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1076;
        2|^16 = 1076*60+976 &
          976*976*976*976*2|^3 = 1076*(82136*82137+71470)+456 by Lm16,Lm3;
        then 2|^(16*4+3),456 are_congruent_mod 1076 &
          456*456*456*456*2|^1 = 1076*80366877+540 by Lem1;
        then 2|^(67*4+1),540 are_congruent_mod 1076 &
          540*540*540*540*2|^0 = 1076*79024684+16 by LemS,Lm1;
        then 2|^(269*4+0),16 are_congruent_mod 1076 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1077;
        2|^16 = 1077*60+916 &
          916*916*916*916*2|^3 = 1077*(14923*350429)+629 by Lm16,Lm3;
        then 2|^(16*4+3),629 are_congruent_mod 1077 &
          629*629*629*629*2|^1 = 1077*290681152+1058 by Lem1;
        then 2|^(67*4+1),1058 are_congruent_mod 1077 &
          1058*1058*1058*1058*2|^1 = 1077*(41458*56124)+8 by LemS;
        then 2|^(269*4+1),8 are_congruent_mod 1077 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1078;
        2|^16 = 1078*60+856 &
          856*856*856*856*2|^3 = 1078*(3115*1279111)+898 by Lm16,Lm3;
        then 2|^(16*4+3),898 are_congruent_mod 1078 &
          898*898*898*898*2|^1 = 1078*1206470150+732 by Lem1;
        then 2|^(67*4+1),732 are_congruent_mod 1078 &
          732*732*732*732*2|^2 = 1078*1065333428+520 by LemS,Lm2;
        then 2|^(269*4+2),520 are_congruent_mod 1078 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1079;
        2|^16 = 1079*60+796 &
          796*796*796*796*2|^3 = 1079*(2907*1023943)+869 by Lm16,Lm3;
        then 2|^(16*4+3),869 are_congruent_mod 1079 &
          869*869*869*869*2|^1 = 1079*1057030835+877 by Lem1;
        then 2|^(67*4+1),877 are_congruent_mod 1079 &
          877*877*877*877*2|^3 = 1079*(55599*78886)+722 by LemS,Lm3;
        then 2|^(269*4+3),722 are_congruent_mod 1079 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1080;
        2|^16 = 1080*60+736 &
          736*736*736*736*2|^3 = 1080*(14*155256379)+848 by Lm16,Lm3;
        then 2|^(16*4+3),848 are_congruent_mod 1080 &
          848*848*848*848*2|^2 = 1080*1915224306+784 by Lem1,Lm2;
        then 2|^(67*4+2),784 are_congruent_mod 1080 &
          784*784*784*784*2|^0 = 1080*349816665+136 by LemS,Lm1;
        then 2|^(270*4+0),136 are_congruent_mod 1080 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1081;
        2|^16 = 1081*60+676 &
          676*676*676*676*2|^3 = 1081*1545436185+623 by Lm16,Lm3;
        then 2|^(16*4+3),623 are_congruent_mod 1081 &
          623*623*623*623*2|^2 = 1081*557425053+271 by Lem1,Lm2;
        then 2|^(67*4+2),271 are_congruent_mod 1081 &
          271*271*271*271*2|^1 = 1081*9978872+330 by LemS;
        then 2|^(270*4+1),330 are_congruent_mod 1081 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1082;
        2|^16 = 1082*60+616 &
          616*616*616*616*2|^3 = 1082*1064597825+838 by Lm16,Lm3;
        then 2|^(16*4+3),838 are_congruent_mod 1082 &
          838*838*838*838*2|^2 = 1082*1823092922+540 by Lem1,Lm2;
        then 2|^(67*4+2),540 are_congruent_mod 1082 &
          540*540*540*540*2|^2 = 1082*314345878+4 by LemS;
        then 2|^(270*4+2),4 are_congruent_mod 1082 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1083;
        2|^16 = 1083*60+556 &
          556*556*556*556*2|^3 = 1083*705928469+41 by Lm16,Lm3;
        then 2|^(16*4+3),41 are_congruent_mod 1083 &
          41*41*41*41*2|^2 = 1083*10436+856 by Lem1,Lm2;
        then 2|^(67*4+2),856 are_congruent_mod 1083 &
          856*856*856*856*2|^3 = 1083*(275*14421947)+293 by LemS,Lm3;
        then 2|^(270*4+3),293 are_congruent_mod 1083 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1084;
        2|^16 = 1084*60+496 &
          496*496*496*496*2|^3 = 1084*446670643+1036 by Lm16,Lm3;
        then 2|^(16*4+3),1036 are_congruent_mod 1084 &
          1036*1036*1036*1036*2|^3 = 1084*(8*1062697697)+544 by Lem1;
        then 2|^(67*4+3),544 are_congruent_mod 1084 &
          544*544*544*544*2|^0 = 1084*80791620+16 by LemS,Lm1;
        then 2|^(271*4+0),16 are_congruent_mod 1084 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1085;
        2|^16 = 1085*60+436 &
          436*436*436*436*2|^3 = 1085*266444160+128 by Lm16,Lm3;
        then 2|^(16*4+3),128 are_congruent_mod 1085 &
          128*128*128*128*2|^3 = 1085*1979247+653 by Lem1;
        then 2|^(67*4+3),653 are_congruent_mod 1085 &
          653*653*653*653*2|^1 = 1085*335160618+32 by LemS;
        then 2|^(271*4+1),32 are_congruent_mod 1085 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1086;
        2|^16 = 1086*60+376 &
          376*376*376*376*2|^3 = 1086*147235162+1076 by Lm16,Lm3;
        then 2|^(16*4+3),1076 are_congruent_mod 1086 &
          1076*1076*1076*1076*2|^3 = 1086*(43031*229471)+722 by Lem1;
        then 2|^(67*4+3),722 are_congruent_mod 1086 &
          722*722*722*722*2|^2 = 1086*1000872960+64 by LemS,Lm2;
        then 2|^(271*4+2),64 are_congruent_mod 1086 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1087;
        hence thesis by XPRIMES1:1087;
      end;
      suppose A: n = 1088;
        2|^4 = 1088*0+16 & 16*16*16*16*2|^1 = 1088*120+512 by Lm4;
        then 2|^(4*4+1),512 are_congruent_mod 1088 &
          512*512*512*512*2|^0 = 1088*63161283+832 by Lem1,Lm1;
        then 2|^(17*4+0),832 are_congruent_mod 1088 &
          832*832*832*832*2|^0 = 1088*440417340+256 by LemS;
        then 2|^(68*4+0),256 are_congruent_mod 1088 &
          256*256*256*256*2|^0 = 1088*3947580+256 by LemS;
        then 2|^(272*4+0),256 are_congruent_mod 1088 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1089;
        2|^4 = 1089*0+16 & 16*16*16*16*2|^1 = 1089*120+392 by Lm4;
        then 2|^(4*4+1),392 are_congruent_mod 1089 &
          392*392*392*392*2|^0 = 1089*21682851+157 by Lem1,Lm1;
        then 2|^(17*4+0),157 are_congruent_mod 1089 &
          157*157*157*157*2|^0 = 1089*557918+499 by LemS;
        then 2|^(68*4+0),499 are_congruent_mod 1089 &
          499*499*499*499*2|^1 = 1089*113868683+215 by LemS;
        then 2|^(272*4+1),215 are_congruent_mod 1089 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1090;
        2|^4 = 1090*0+16 & 16*16*16*16*2|^1 = 1090*120+272 by Lm4;
        then 2|^(4*4+1),272 are_congruent_mod 1090 &
          272*272*272*272*2|^0 = 1090*5021680+1056 by Lem1,Lm1;
        then 2|^(17*4+0),1056 are_congruent_mod 1090 &
          1056*1056*1056*1056*2|^0 = 1090*1140851649+1086 by LemS;
        then 2|^(68*4+0),1086 are_congruent_mod 1090 &
          1086*1086*1086*1086*2|^2 = 1090*(63752*80068)+1024 by LemS,Lm2;
        then 2|^(272*4+2),1024 are_congruent_mod 1090 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1091;
        hence thesis by XPRIMES1:1091;
      end;
      suppose A: n = 1092;
        2|^4 = 1092*0+16 & 16*16*16*16*2|^1 = 1092*120+32 by Lm4;
        then 2|^(4*4+1),32 are_congruent_mod 1092 &
          32*32*32*32*2|^0 = 1092*960+256 by Lem1,Lm1;
        then 2|^(17*4+0),256 are_congruent_mod 1092 &
          256*256*256*256*2|^1 = 1092*7866240+512 by LemS;
        then 2|^(68*4+1),512 are_congruent_mod 1092 &
          512*512*512*512*2|^0 = 1092*62929923+820 by LemS,Lm1;
        then 2|^(273*4+0),820 are_congruent_mod 1092 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1093;
        hence thesis by XPRIMES1:1093;
      end;
      suppose A: n = 1094;
        2|^4 = 1094*0+16 & 16*16*16*16*2|^1 = 1094*119+886 by Lm4;
        then 2|^(4*4+1),886 are_congruent_mod 1094 &
          886*886*886*886*2|^0 = 1094*563271224+960 by Lem1,Lm1;
        then 2|^(17*4+0),960 are_congruent_mod 1094 &
          960*960*960*960*2|^1 = 1094*1552735941+546 by LemS;
        then 2|^(68*4+1),546 are_congruent_mod 1094 &
          546*546*546*546*2|^2 = 1094*324947530+4 by LemS,Lm2;
        then 2|^(273*4+2),4 are_congruent_mod 1094 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1095;
        2|^4 = 1095*0+16 & 16*16*16*16*2|^1 = 1095*119+767 by Lm4;
        then 2|^(4*4+1),767 are_congruent_mod 1095 &
          767*767*767*767*2|^0 = 1095*316058399+616 by Lem1,Lm1;
        then 2|^(17*4+0),616 are_congruent_mod 1095 &
          616*616*616*616*2|^1 = 1095*262989691+227 by LemS;
        then 2|^(68*4+1),227 are_congruent_mod 1095 &
          227*227*227*227*2|^3 = 1095*19398997+1013 by LemS,Lm3;
        then 2|^(273*4+3),1013 are_congruent_mod 1095 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1096;
        2|^4 = 1096*0+16 & 16*16*16*16*2|^1 = 1096*119+648 by Lm4;
        then 2|^(4*4+1),648 are_congruent_mod 1096 &
          648*648*648*648*2|^0 = 1096*160875336+960 by Lem1,Lm1;
        then 2|^(17*4+0),960 are_congruent_mod 1096 &
          960*960*960*960*2|^2 = 1096*(3*1033268321)+552 by LemS,Lm2;
        then 2|^(68*4+2),552 are_congruent_mod 1096 &
          552*552*552*552*2|^0 = 1096*84712160+256 by LemS,Lm1;
        then 2|^(274*4+0),256 are_congruent_mod 1096 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1097;
        hence thesis by XPRIMES1:1097;
      end;
      suppose A: n = 1098;
        2|^4 = 1098*0+16 & 16*16*16*16*2|^1 = 1098*119+410 by Lm4;
        then 2|^(4*4+1),410 are_congruent_mod 1098 &
          410*410*410*410*2|^0 = 1098*25735528+256 by Lem1,Lm1;
        then 2|^(17*4+0),256 are_congruent_mod 1098 &
          256*256*256*256*2|^2 = 1098*15646511+106 by LemS,Lm2;
        then 2|^(68*4+2),106 are_congruent_mod 1098 &
          106*106*106*106*2|^2 = 1098*459918+820 by LemS;
        then 2|^(274*4+2),820 are_congruent_mod 1098 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1099;
        2|^4 = 1099*0+16 & 16*16*16*16*2|^1 = 1099*119+291 by Lm4;
        then 2|^(4*4+1),291 are_congruent_mod 1099 &
          291*291*291*291*2|^0 = 1099*6524906+67 by Lem1,Lm1;
        then 2|^(17*4+0),67 are_congruent_mod 1099 &
          67*67*67*67*2|^2 = 1099*73343+527 by LemS,Lm2;
        then 2|^(68*4+2),527 are_congruent_mod 1099 &
          527*527*527*527*2|^3 = 1099*561480600+128 by LemS,Lm3;
        then 2|^(274*4+3),128 are_congruent_mod 1099 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1100;
        2|^4 = 1100*0+16 & 16*16*16*16*2|^1 = 1100*119+172 by Lm4;
        then 2|^(4*4+1),172 are_congruent_mod 1100 &
          172*172*172*172*2|^0 = 1100*795648+256 by Lem1,Lm1;
        then 2|^(17*4+0),256 are_congruent_mod 1100 &
          256*256*256*256*2|^3 = 1100*31236125+868 by LemS,Lm3;
        then 2|^(68*4+3),868 are_congruent_mod 1100 &
          868*868*868*868*2|^0 = 1100*516043385+276 by LemS,Lm1;
        then 2|^(275*4+0),276 are_congruent_mod 1100 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1101;
        2|^4 = 1101*0+16 & 16*16*16*16*2|^1 = 1101*119+53 by Lm4;
        then 2|^(4*4+1),53 are_congruent_mod 1101 &
          53*53*53*53*2|^0 = 1101*7166+715 by Lem1,Lm1;
        then 2|^(17*4+0),715 are_congruent_mod 1101 &
          715*715*715*715*2|^3 = 1101*1899008178+1022 by LemS,Lm3;
        then 2|^(68*4+3),1022 are_congruent_mod 1101 &
          1022*1022*1022*1022*2|^1 = 1101*1981738104+8 by LemS;
        then 2|^(275*4+1),8 are_congruent_mod 1101 by LemS;
        hence thesis by A,LemF;
      end;
      suppose A: n = 1102;
        2|^4 = 1102*0+16 & 16*16*16*16*2|^1 = 1102*118+1036 by Lm4;
        then 2|^(4*4+1),1036 are_congruent_mod 1102 &
          1036*1036*1036*1036*2|^0 = 1102*1045339658+500 by Lem1,Lm1;
        then 2|^(17*4+0),500 are_congruent_mod 1102 &
          500*500*500*500*2|^3 = 1102*453720508+184 by LemS,Lm3;
        then 2|^(68*4+3),184 are_congruent_mod 1102 &
          184*184*184*184*2|^2 = 1102*4160539+966 by LemS,Lm2;
        then 2|^(275*4+2),966 are_congruent_mod 1102 by LemS;
        hence thesis by A,LemF;
      end;
      suppose n = 1103;
        hence thesis by XPRIMES1:1103;
      end;
      suppose A: n = 1104;
        2|^4 = 1104*0+16 & 16*16*16*16*2|^1 = 1104*118+800 by Lm4;
        then 2|^(4*4+1),800 are_congruent_mod 1104 &
          800*800*800*800*2|^1 = 1104*742028985+560 by Lem1;
        then 2|^(17*4+1),560 are_congruent_mod 1104 &
          560*560*560*560*2|^0 = 1104*89080579+784 by LemS,Lm1;
        then 2|^(69*4+0),784 are_congruent_mod 1104 &
          784*784*784*784*2|^0 = 1104*342211955+16 by LemS;
        then 2|^(276*4+0),16 are_congruent_mod 1104 by LemS;
        hence thesis by A,LemF;
      end;
    end;
    suppose 1104 < n & n <= 1105;
      then 1104+1 <= n & n <= 1105 by NAT_1:13;
      then n = 1105 by XXREAL_0:1;
      hence thesis by A5,ENUMSET1:def 2;
    end;
  end;
