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
  holds C is ManySortedSubset of [|the Sorts of A, the Sorts of A|]
proof
  let A be non-empty MSAlgebra over S, C be MSCongruence of A;
  set SF = the Sorts of A;
  let i be object such that
A1: i in the carrier of S;
  C.i is Relation of SF.i, SF.i by A1,MSUALG_4:def 1;
  then C.i c= [:SF.i, SF.i:];
  hence thesis by A1,PBOOLE:def 16;
end;
