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 Th52:
  for W,W1,W2 st W + W1 = ID(M) & W + W2 = ID(M) holds W1 = W2
proof
  let W,W1,W2 such that
A1: W + W1 = ID(M) & W + W2 = ID(M);
  reconsider x = W,y1 = W1,y2 = W2 as Vector of M by Th48;
  x + y1 = W + W2 by A1,Def13
    .= x + y2 by Def13;
  hence thesis by Th47;
end;
