
theorem Lm1:
for R be non empty Abelian addLoopStr, n be Nat
for u,v being Tuple of n,(the carrier of R) holds u + v = v + u
proof
let R be non empty Abelian addLoopStr,
    n be Nat, u,v be Tuple of n,(the carrier of R);
set w1 = u + v, w2 = v + u;
reconsider u1 = u, v1 = v as Element of n-tuples_on the carrier of R
   by FINSEQ_2:131;
now let i be Nat;
  assume AS: i in Seg n;
  reconsider a = u/.i, b = v/.i as Element of R;
  J: dom v = Seg len v1 by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:133;
  dom u = Seg len u1 by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:133; then
  H: a = u1.i & b = v1.i by J,AS,PARTFUN1:def 6; then
  w1.i = a + b by AS,FVSUM_1:18 .= w2.i by H,AS,FVSUM_1:18;
  hence w1.i = w2.i;
  end;
hence thesis by FINSEQ_2:119;
end;
