reserve a, I for set,
  S for non empty non void ManySortedSign;
reserve A, M for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem
  for A being non-empty MSAlgebra over S for C being MSCongruence of A
st C = [|the Sorts of A, the Sorts of A|] holds the Sorts of QuotMSAlg (A,C) =
  {the Sorts of A}
proof
  let A be non-empty MSAlgebra over S, C be MSCongruence of A such that
A1: C = [|the Sorts of A, the Sorts of A|];
  now
    let i be object;
    assume i in the carrier of S;
    then reconsider s = i as Element of S;
A2: C.s = [:(the Sorts of A).s, (the Sorts of A).s:] by A1,PBOOLE:def 16
      .= nabla (the Sorts of A).s by EQREL_1:def 1;
    thus (the Sorts of QuotMSAlg (A,C)).i = Class (C.s) by MSUALG_4:def 6
      .= {(the Sorts of A).s} by A2,Th4
      .= {the Sorts of A}.i by PZFMISC1:def 1;
  end;
  hence thesis;
end;
