reserve X for set, x,y,z for object,
  k,l,n for Nat,
  r for Real;
reserve i,i0,i1,i2,i3,i4,i5,i8,i9,j for Integer;
reserve r1,p,p1,g,g1,g2 for Real,
  Y for Subset of REAL;
reserve r, s for Real;
reserve i for Integer,
  a, b, r, s for Real;

theorem ::: INT_4:12
  for i being Integer holds 0 = 0 mod i
proof   
  let i be Integer;
  per cases;
  suppose
    i=0;
    hence thesis by Def10;
  end;   
  suppose
A1: i<>0;
    0 div i = 0;
    then 0 mod i = 0-i*(0 qua Nat) by A1,Def10;
    hence thesis;
  end;
end;  
