
theorem Th7:
  for I being non empty set for A being 1-sorted-yielding
  ManySortedSet of I for L being ManySortedSubset of Carrier A for i being
  Element of I for S being Subset of A.i holds L+*(i,S) is ManySortedSubset of
  Carrier A
proof
  let I be non empty set;
  let A be 1-sorted-yielding ManySortedSet of I;
  let L be ManySortedSubset of Carrier A;
  let i be Element of I;
  let S be Subset of A.i;
A1: L c= Carrier A by PBOOLE:def 18;
A2: dom L = I by PARTFUN1:def 2;
  L+*(i,S) c= Carrier A
  proof
    let a be object;
    assume a in I;
    then reconsider b=a as Element of I;
    per cases;
    suppose
A3:   a = i;
      then L+*(i,S).b = S by A2,FUNCT_7:31;
      then L+*(i,S).b c= the carrier of A.b by A3;
      hence thesis by YELLOW_6:2;
    end;
    suppose
      a <> i;
      then L+*(i,S).a = L.b by FUNCT_7:32;
      hence thesis by A1;
    end;
  end;
  hence thesis by PBOOLE:def 18;
end;
