
theorem Th25:
  for I being non empty set for A being PLS-yielding ManySortedSet
  of I for P being ManySortedSet of I st P is Point of Segre_Product A for i
  being Element of I for p being Point of A.i holds P+*(i,p) is Point of
  Segre_Product A
proof
  let I be non empty set;
  let A be PLS-yielding ManySortedSet of I;
  let P be ManySortedSet of I such that
A1: P is Point of Segre_Product A;
  let j be Element of I;
  let p be Point of A.j;
A2: for i be object st i in dom Carrier A holds P+*(j,p).i in (Carrier A).i
  proof
    let i be object;
    assume
A3: i in dom Carrier A;
    then i in I by PARTFUN1:def 2;
    then
A4: i in dom P by PARTFUN1:def 2;
    per cases;
    suppose
      i <> j;
      then P+*(j,p).i = P.i by FUNCT_7:32;
      hence thesis by A1,A3,CARD_3:9;
    end;
    suppose
A5:   i = j;
A6:   p in the carrier of A.j;
      P+*(j,p).i = p by A4,A5,FUNCT_7:31;
      hence thesis by A5,A6,YELLOW_6:2;
    end;
  end;
  dom (P+*(j,p)) = I by PARTFUN1:def 2
    .= dom Carrier A by PARTFUN1:def 2;
  hence thesis by A2,CARD_3:9;
end;
