reserve a,b,p,k,l,m,n,s,h,i,j,t,i1,i2 for natural Number;

theorem Th64:
  for n,a,b being Integer holds
  (n <> 0 & a mod n = b mod n implies a,b are_congruent_mod n) &
  (a,b are_congruent_mod n implies a mod n = b mod n)
proof
  let n,a,b be Integer;
  hereby
    assume
A1: n <> 0;
    assume a mod n = b mod n;
    then a - (a div n) * n = b mod n by A1,INT_1:def 10;
    then a - (a div n) * n = b - (b div n) * n by A1,INT_1:def 10;
    then a - b = (-(b div n) + (a div n)) * n;
    then n divides (a-b) by INT_1:def 3;
    hence a,b are_congruent_mod n by INT_2:15;
  end;
  assume a,b are_congruent_mod n;
  then n divides (a-b) by INT_2:15;
  then consider k being Integer such that
A2: n * k = a - b by INT_1:def 3;
  a = n * k + b by A2;
  hence thesis by Th61;
end;
