reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;

theorem Th11:
  i1,i2 are_congruent_mod i implies i1*i3,i2*i3 are_congruent_mod i
proof
  assume i1,i2 are_congruent_mod i;
  then i divides (i1-i2);
  then i divides (i1-i2)*i3 by INT_2:2;
  then i divides (i1*i3-i2*i3);
  hence thesis;
end;
