reserve S for OrderSortedSign;
reserve S for OrderSortedSign,
  X for ManySortedSet of S,
  o for OperSymbol of S ,
  b for Element of ([:the carrier' of S,{the carrier of S}:] \/ Union (coprod X
  ))*;
reserve x for set;

theorem
  for S be locally_directed regular monotone OrderSortedSign, X be
  non-empty ManySortedSet of S holds LCongruence(X) = PTCongruence(X)
proof
  let S be locally_directed regular monotone OrderSortedSign, X be non-empty
  ManySortedSet of S;
A1: PTCongruence(X) c= LCongruence(X) by Th45;
  LCongruence(X) c= PTCongruence(X) by Def17;
  hence thesis by A1,PBOOLE:146;
end;
