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 Th51:
  for W ex T st W + T = ID(M)
proof
  let W;
  reconsider x = W as Vector of M by Th48;
  consider y being Vector of M such that
A1: x + y = ID(M) by Th45;
  reconsider T = y as Element of setvect(M) by Th48;
  x + y = W + T by Def13;
  hence thesis by A1;
end;
