
theorem Th23:
  for I being non empty set for A being Segre-like non
  trivial-yielding ManySortedSet of I for x,y being ManySortedSet of I st x in
  product A & y in product A
for i being object st i <> indx(A) holds x.i = y.i
proof
  let I be non empty set;
  let A be Segre-like non trivial-yielding ManySortedSet of I;
  let x,y be ManySortedSet of I such that
A1: x in product A and
A2: y in product A;
  let i be object;
A3: dom A = I by PARTFUN1:def 2;
  assume
A4: i <> indx(A);
  per cases;
  suppose
A5: not i in I;
    then
A6: not i in dom y by PARTFUN1:def 2;
    not i in dom x by A5,PARTFUN1:def 2;
    hence x.i = {} by FUNCT_1:def 2
      .= y.i by A6,FUNCT_1:def 2;
  end;
  suppose
    i in I;
    then reconsider ii=i as Element of I;
    consider j being Element of I such that
A7: for k being Element of I st k<>j holds A.k is 1-element by Def20;
    now
      assume j <> indx(A);
      then A.indx(A) is 1-element by A7;
      hence contradiction by Def21;
    end;
    then A.ii is 1-element by A4,A7;
    then consider o being object such that
A8: A.i = {o} by ZFMISC_1:131;
A9: x.ii in A.ii by A1,A3,CARD_3:9;
    y.ii in A.ii by A2,A3,CARD_3:9;
    hence y.i = o by A8,TARSKI:def 1
      .= x.i by A8,A9,TARSKI:def 1;
  end;
end;
