reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th19:
  for i,j,k being Integer holds
  i > 0 & j,k are_congruent_mod i implies (i divides j iff i divides k)
  proof
    let i,j,k be Integer;
    assume
A1: i > 0;
    assume j,k are_congruent_mod i;
    then
A2: j mod i = k mod i by NAT_D:64;
    thus i divides j implies i divides k
    proof
      assume i divides j;
      then j mod i = 0 by A1,INT_1:62;
      hence thesis by A1,A2,INT_1:62;
    end;
    assume i divides k;
    then k mod i = 0 by A1,INT_1:62;
    hence thesis by A1,A2,INT_1:62;
  end;
