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 Th65:
  7 divides 10|^(6*k+4) + 3
  proof
    10000 = 1428*7+4;
    then
A1: 10|^4,4 are_congruent_mod 7 by Lm2062;
A2: 7,0 are_congruent_mod 7;
    per cases;
    suppose
A3:   k = 0;
      10|^4+3,4+3 are_congruent_mod 7 by A1;
      then 10|^4+3,0 are_congruent_mod 7 by A2,INT_1:15;
      hence thesis by A3;
    end;
    suppose
A4:   k <> 0;
      1*7+3,7 are_coprime by Th64,XPRIMES1:3,7,INT_2:30;
      then
A5:   (10 |^ Euler 7) mod 7 = 1 by EULER_2:18;
      Euler 7 = 7-1 by EULER_1:20,XPRIMES1:7;
      then
A6:   10|^6|^k mod 7 = 1 by A5,A4,NEWTON05:15;
A7:   1 mod 7 = 1 by NAT_D:24;
      10|^6|^k = 10|^(6*k) by NEWTON:9;
      then 10|^(6*k),1 are_congruent_mod 7 by A6,A7,NAT_D:64;
      then
A8:   10|^(6*k)*10|^4,1*4 are_congruent_mod 7 by A1,INT_1:18;
      10|^(6*k)*10|^4 = 10|^(6*k+4) by NEWTON:8;
      then 10|^(6*k+4)+3,4+3 are_congruent_mod 7 by A8;
      then 10|^(6*k+4)+3,0 are_congruent_mod 7 by A2,INT_1:15;
      hence thesis;
    end;
  end;
