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 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;
