theorem Th10:
  -(F1 + F2) = -F1 + -F2
proof
A1: dom -(F1 + F2) = dom(F1+F2) by VALUED_1:8;
A2: dom(F1+F2) = dom F1 /\ dom F2 by VALUED_1:def 1;
A3: dom (-F1 + -F2) = dom (-F1) /\ dom -F2 by VALUED_1:def 1
    .= dom F1 /\ dom -F2 by VALUED_1:8
    .= dom F1 /\ dom F2 by VALUED_1:8;
  now
    let i;
    assume A4: i in dom -(F1+F2);
    thus (-(F1 + F2)).i = -((F1+F2).i) by VALUED_1:8
    .= -(F1.i+F2.i) by A1,A4,VALUED_1:def 1
    .= -(F1.i)+-(F2.i)
    .= -(F1.i) + (-F2).i by VALUED_1:8
    .= (-F1).i + (-F2).i by VALUED_1:8
    .= (-F1 + -F2).i by A1,A2,A3,A4,VALUED_1:def 1;
  end;
  hence thesis by A2,A3,FINSEQ_1:13,VALUED_1:8;
end;
