
theorem Th11:
  for I being non empty set for A being non-Empty
1-sorted-yielding ManySortedSet of I for B being Element of Carrier A holds {B
  } is ManySortedSubset of Carrier A
proof
  let I be non empty set;
  let A be non-Empty 1-sorted-yielding ManySortedSet of I;
  let B be Element of Carrier A;
  {B} c= Carrier A
  proof
    let i be object;
    assume
A1: i in I;
    then reconsider j=i as Element of I;
    j in dom A by A1,PARTFUN1:def 2;
    then A.j in rng A by FUNCT_1:def 3;
    then
A2: A.j is non empty by YELLOW_6:def 2;
    B.j is Element of (Carrier A).j by PBOOLE:def 14;
    then B.j is Element of A.j by YELLOW_6:2;
    then {B.j} c= the carrier of A.j by A2,ZFMISC_1:31;
    then {B}.j c= the carrier of A.j by PZFMISC1:def 1;
    hence thesis by YELLOW_6:2;
  end;
  hence thesis by PBOOLE:def 18;
end;
