
theorem Th12:
  for I being non empty set for A being Segre-like non
  trivial-yielding ManySortedSet of I for i being Element of I st i <> indx(A)
  holds A.i is 1-element
proof
  let I be non empty set;
  let A be Segre-like non trivial-yielding ManySortedSet of I;
  let i be Element of I;
  consider j being Element of I such that
A1: for k being Element of I st k<>j holds A.k is 1-element by Def20;
A2: now
    assume indx(A) <> j;
    then A.indx(A) is 1-element by A1;
    hence contradiction by Def21;
  end;
  assume i <> indx(A);
  hence thesis by A1,A2;
end;
