
theorem Th1:
  for K being non empty addLoopStr, p1,p2 be FinSequence of the
  carrier of K st dom p1 = dom p2 holds dom(p1+p2) = dom p1
proof
  let K be non empty addLoopStr, p1,p2 be FinSequence of the carrier of K;
  assume
A1: dom p1 = dom p2;
A2: rng <:p1,p2:> c= [:rng p1,rng p2:] & [:rng p1,rng p2:] c= [:the carrier
  of K,the carrier of K:] by FUNCT_3:51,ZFMISC_1:96;
A3: [:the carrier of K,the carrier of K:] = dom (the addF of K) by
FUNCT_2:def 1;
  thus dom (p1+p2) = dom ((the addF of K).:(p1,p2)) by FVSUM_1:def 3
    .= dom <:p1,p2:> by A2,A3,RELAT_1:27,XBOOLE_1:1
    .= dom p1 /\ dom p2 by FUNCT_3:def 7
    .= dom p1 by A1;
end;
