reserve MS for non empty MidStr;
reserve a, b for Element of MS;
reserve M for MidSp;
reserve a,b,c,d,a9,b9,c9,d9,x,y,x9 for Element of M;
reserve p,q,r,p9,q9 for Element of [:the carrier of M,the carrier of M:];
reserve u,v,w,u9,w9 for Vector of M;
reserve X for Subset of [:the carrier of M,the carrier of M:];
reserve x for set;
reserve u1,v1,w1,W,W1,W2,T for Element of setvect(M);

theorem Th50:
  (u1 + v1) + w1 = u1 + (v1 + w1)
proof
  reconsider u = u1, v = v1, w = w1 as Vector of M by Th48;
A1: v1 + w1 = v + w by Def13;
  u1 + v1 = u + v by Def13;
  hence (u1 + v1) + w1 = (u + v) + w by Def13
    .= u + (v + w) by Th43
    .= u1 + (v1 + w1) by A1,Def13;
end;
