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 Th7:
  for i3 st i3 <> 0 holds i1 divides i2 iff i1*i3 divides i2*i3
proof
  let i3;
  assume
A1: i3<>0;
  thus i1 divides i2 implies i1*i3 divides i2*i3
  proof
    assume i1 divides i2;
    then consider i4 being Integer such that
A2: i2=i1*i4;
    i2*i3=(i1*i3)*i4 by A2;
    hence thesis;
  end;
  assume (i1*i3) divides (i2*i3);
  then consider i5 being Integer such that
A3: (i2*i3)=(i1*i3)*i5;
  i2*i3=(i1*i5)*i3 by A3;
  then i2=i1*i5 by A1,XCMPLX_1:5;
  hence thesis;
end;
