reserve N for Nat;
reserve n,m,n1,n2 for Nat;
reserve q,r,r1,r2 for Real;
reserve x,y for set;
reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N;
reserve seq,seq1,seq2,seq3,seq9 for Real_Sequence of N;

theorem Th10:
  (seq1 + seq2) + seq3 = seq1 + (seq2 + seq3)
proof
  let n be Element of NAT;
  thus ((seq1+seq2)+seq3).n = (seq1+seq2).n+ seq3.n by Th4
    .= seq1.n+seq2.n+seq3.n by Th4
    .=seq1.n+(seq2.n+seq3.n) by RLVECT_1:def 3
    .=seq1.n+(seq2+seq3).n by Th4
    .=(seq1+(seq2+seq3)).n by Th4;
end;
