reserve a,b,c for Integer;
reserve i,j,k,l for Nat;
reserve n for Nat;
reserve a,b,c,d,a1,b1,a2,b2,k,l for Integer;

theorem Th10:
  (a divides b iff a divides -b) & (a divides b iff -a divides b) &
  (a divides b iff -a divides -b) & (a divides -b iff -a divides b)
proof
A1: a divides b implies a divides -b
  proof
    assume
A2: a divides b;
    b divides -b by Lm1;
    hence thesis by A2,Lm2;
  end;
A3: a divides -b implies a divides b
  proof
    assume
A4: a divides -b;
    -b divides b by Lm1;
    hence thesis by A4,Lm2;
  end;
  hence a divides b iff a divides -b by A1;
A5: -a divides b implies a divides b
  proof
    assume
A6: -a divides b;
    a divides -a by Lm1;
    hence thesis by A6,Lm2;
  end;
A7: -a divides -b implies a divides b
  proof
    assume
A8: -a divides -b;
    -b divides b by Lm1;
    hence thesis by A5,A8,Lm2;
  end;
A9: a divides b implies -a divides b
  proof
    assume
A10: a divides b;
    -a divides a by Lm1;
    hence thesis by A10,Lm2;
  end;
  hence a divides b iff -a divides b by A5;
  a divides b implies -a divides -b
  proof
    assume
A11: a divides b;
    -a divides a by Lm1;
    hence thesis by A1,A11,Lm2;
  end;
  hence a divides b iff -a divides -b by A7;
  thus thesis by A1,A3,A9,A5;
end;
