reserve a,b,c,k,k9,m,n,n9,p,p9 for Nat;
reserve i,i9 for Integer;

theorem Th7:
  m divides n or k = 0 iff k*m divides k*n
proof
  hereby
    assume
A1: m divides n or k = 0;
    per cases by A1;
    suppose
      m divides n;
      then consider k9 be Nat such that
A2:   n = m*k9 by NAT_D:def 3;
      k*n = (k*m)*k9 by A2;
      hence k*m divides k*n;
    end;
    suppose
      k = 0;
      hence k*m divides k*n;
    end;
  end;
  assume
A3: k*m divides k*n;
  now
    consider k9 be Nat such that
A4: k*n = k*m*k9 by A3,NAT_D:def 3;
    assume
A5: k <> 0;
    k*n = k*(m*k9) by A4;
    then n = m*k9 by A5,XCMPLX_1:5;
    hence m divides n;
  end;
  hence thesis;
end;
