
theorem Th31:
  for M being non empty set, i being Element of M, J being
  TopStruct-yielding non-Empty ManySortedSet of M, x being Point of product J
  holds pi(Cl {x}, i) = Cl {x.i}
proof
  let M be non empty set, i be Element of M, J be TopStruct-yielding non-Empty
  ManySortedSet of M, x be Point of product J;
  consider f being object such that
A1: f in Cl {x} by XBOOLE_0:def 1;
A2: f in the carrier of product J by A1;
  thus pi(Cl {x}, i) c= Cl {x.i}
  proof
    let a be object;
    assume a in pi(Cl {x}, i);
    then ex f being Function st f in Cl {x} & a = f.i by CARD_3:def 6;
    hence thesis by Th29;
  end;
  reconsider f as Element of product J by A1;
  let a be object;
  set y = f +* (i .--> a);
A3: dom Carrier J = M by PARTFUN1:def 2;
A4: f in product Carrier J by A2,WAYBEL18:def 3;
  then
A5: dom f = M by A3,CARD_3:9;
  assume
A7: a in Cl {x.i};
A8: for m being object st m in dom Carrier J holds y.m in (Carrier J).m
  proof
    let m be object;
    assume
A9: m in dom Carrier J;
    then
A10: ex R being 1-sorted st R = J.m & Carrier J.m = the carrier of R by
PRALG_1:def 15;
    per cases;
    suppose
A11:  m = i;
      then y.m = a by FUNCT_7:94;
      hence thesis by A7,A10,A11;
    end;
    suppose
      m <> i;
      then not m in dom (i .--> a) by TARSKI:def 1;
      then y.m = f.m by FUNCT_4:11;
      hence thesis by A4,A9,CARD_3:9;
    end;
  end;
  dom y = dom f \/ dom (i .--> a) by FUNCT_4:def 1
    .= M \/ {i} by A5
    .= M by ZFMISC_1:40;
  then y in product Carrier J by A3,A8,CARD_3:9;
  then reconsider y = f +* (i .--> a) as Element of product J by WAYBEL18:def 3
;
  for m being Element of M holds y.m in Cl {x.m}
  proof
    let m be Element of M;
    per cases;
    suppose
      m = i;
      hence thesis by A7,FUNCT_7:94;
    end;
    suppose
      m <> i;
      then not m in dom (i .--> a) by TARSKI:def 1;
      then y.m = f.m by FUNCT_4:11;
      hence thesis by A1,Th29;
    end;
  end;
  then
A12: f +* (i .--> a) in Cl {x} by Th29;
  (f +* (i .--> a)).i = a by FUNCT_7:94;
  hence thesis by A12,CARD_3:def 6;
end;
