
theorem Th1:
  for L be add-associative right_zeroed right_complementable non
empty addLoopStr for p be FinSequence of the carrier of L st for i be Element
  of NAT st i in dom p holds p.i = 0.L holds Sum p = 0.L
proof
  let L be add-associative right_zeroed right_complementable non empty
  addLoopStr;
  let p be FinSequence of the carrier of L;
  assume
A1: for k be Element of NAT st k in dom p holds p.k = 0.L;
  now
    let k be Nat;
    assume
A2: k in dom p;
    hence p/.k = p.k by PARTFUN1:def 6
      .= 0.L by A1,A2;
  end;
  hence thesis by MATRLIN:11;
end;
