theorem Th10:
  i1,i2 are_congruent_mod i3 implies i1*k,i2*k are_congruent_mod i3*k
proof
  assume i1,i2 are_congruent_mod i3;
  then consider i4 being Integer such that
A1: i3*i4=i1-i2;
  (i3*k)*i4=(i3*i4)*k .=(i1+(-i2))*k by A1
    .=i1*k-i2*k;
  hence thesis;
end;
