
theorem
  for A being non-empty partial UAStr for f being operation of A holds
  f is_exactly_partitable_wrt SmallestPartition the carrier of A
proof
  let A be non-empty partial UAStr;
  set P = SmallestPartition the carrier of A;
  let f be operation of A;
  hereby
    let p be FinSequence of P;
    consider q being FinSequence of the carrier of A such that
A1: product p = {q} by Th12;
    q in dom f & f.q in rng f & rng f c= the carrier of A
    or not q in dom f by FUNCT_1:def 3;
    then consider x being Element of A such that
A2: q in dom f & x = f.q or not q in dom f;
    P = the set of all {z} where z is Element of A by EQREL_1:37;
    then {x} in P;
    then reconsider a = {x} as Element of P;
    take a;
    thus f.:product p c= a
    proof
      let z be object;
      assume z in f.:product p;
      then consider y being object such that
A3:   y in dom f and
A4:   y in product p and
A5:   z = f.y by FUNCT_1:def 6;
      y = q by A1,A4,TARSKI:def 1;
      hence thesis by A2,A3,A5,TARSKI:def 1;
    end;
  end;
  let p be FinSequence of P;
  consider q being FinSequence of the carrier of A such that
A6: product p = {q} by Th12;
  assume product p meets dom f;
  then
A7: ex x being object st x in product p & x in dom f by XBOOLE_0:3;
  let z be object;
  assume z in product p;
  then z = q by A6,TARSKI:def 1;
  hence thesis by A6,A7,TARSKI:def 1;
end;
