
theorem Lm2:
for R be non empty add-associative addLoopStr, n be Nat
for u,v,w being Tuple of n,(the carrier of R) holds (u + v) + w = u + (v + w)
proof
let R be non empty add-associative addLoopStr,
    n be Nat, u,v,w be Tuple of n,(the carrier of R);
set p = u + v, q = v + w;
reconsider u1 = u, v1 = v, w1 = w,
           p1 = p, q2 = q 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, c = w/.i as Element of R;
   J: dom v = Seg len v1 by FINSEQ_1:def 3 .= Seg n by FINSEQ_2:133;
   K: dom w = Seg len w1 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 & c = w1.i by K,J,AS,PARTFUN1:def 6; then
A: p1.i = a + b by AS,FVSUM_1:18;
B: q2.i = b + c by AS,H,FVSUM_1:18;
   thus (p1 + w).i
          = (a + b) + c by AS,A,H,FVSUM_1:18
         .= a + (b + c) by RLVECT_1:def 3
         .= (u + q2).i by AS,B,H,FVSUM_1:18;
   end;
hence thesis by FINSEQ_2:119;
end;
