reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;
reserve O for connected non empty Poset;
reserve R,Q for array of O;
reserve T for non empty array of O;
reserve p,q,r,s for Element of dom T;

theorem Th81:
  for A being 0-based finite array holds last (A^<%x%>) = x
  proof
    let A be 0-based finite array;
    dom <%x%> = 1 by AFINSQ_1:def 4; then
    dom(A^<%x%>) = (dom A)+^1 by ORDINAL4:def 1
    .= succ dom A by ORDINAL2:31; then
    union dom (A^<%x%>) = len A by ORDINAL2:2;
    hence thesis by AFINSQ_1:36;
  end;
