reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem
  7 divides a^2+b^2 implies 7 divides a & 7 divides b
  proof
    assume
A1: 7 divides a^2+b^2;
A2: a^2-a^2 = 0;
A3: b^2-b^2 = 0;
A4: 1 mod 7 = 1 by NAT_D:24;
A5: 2 mod 7 = 2 by NAT_D:24;
A6: 4 mod 7 = 4 by NAT_D:24;
A7: 0,7 are_congruent_mod 7 by INT_1:12;
A8: a^2,-b^2 are_congruent_mod 7 by A1;
A9: b^2,-a^2 are_congruent_mod 7 by A1;
    assume not thesis;
    then per cases;
    suppose not 7 divides a;
      then a^2 mod 7 = 1 or a^2 mod 7 = 2 or a^2 mod 7 = 4 by Th48;
      then
      a^2,1 are_congruent_mod 7 or
      a^2,2 are_congruent_mod 7 or
      a^2,4 are_congruent_mod 7 by A4,A5,A6,NAT_D:64;
      then
      0,-b^2-1 are_congruent_mod 7 or
      0,-b^2-2 are_congruent_mod 7 or
      0,-b^2-4 are_congruent_mod 7 by A2,A8,INT_1:17;
      then
      (-1)*(-b^2),(-1)*1 are_congruent_mod 7 or
      (-1)*(-b^2),(-1)*2 are_congruent_mod 7 or
      (-1)*(-b^2),(-1)*4 are_congruent_mod 7;
      then
      b^2+0,-1+7 are_congruent_mod 7 or
      b^2+0,-2+7 are_congruent_mod 7 or
      b^2+0,-4+7 are_congruent_mod 7 by A7,INT_1:16;
      then b^2 mod 7 = 6 or b^2 mod 7 = 5 or b^2 mod 7 = 3 by NAT_6:12;
      hence thesis by Th49;
    end;
    suppose not 7 divides b;
      then b^2 mod 7 = 1 or b^2 mod 7 = 2 or b^2 mod 7 = 4 by Th48;
      then
      b^2,1 are_congruent_mod 7 or
      b^2,2 are_congruent_mod 7 or
      b^2,4 are_congruent_mod 7 by A4,A5,A6,NAT_D:64;
      then
      0,-a^2-1 are_congruent_mod 7 or
      0,-a^2-2 are_congruent_mod 7 or
      0,-a^2-4 are_congruent_mod 7 by A3,A9,INT_1:17;
      then
      (-1)*(-a^2),(-1)*1 are_congruent_mod 7 or
      (-1)*(-a^2),(-1)*2 are_congruent_mod 7 or
      (-1)*(-a^2),(-1)*4 are_congruent_mod 7;
      then
      a^2+0,-1+7 are_congruent_mod 7 or
      a^2+0,-2+7 are_congruent_mod 7 or
      a^2+0,-4+7 are_congruent_mod 7 by A7,INT_1:16;
      then a^2 mod 7 = 6 or a^2 mod 7 = 5 or a^2 mod 7 = 3 by NAT_6:12;
      hence thesis by Th49;
     end;
  end;
