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 Th32:
  for S be locally_directed OrderSortedSign, X be non-empty
ManySortedSet of S, R be OSCongruence of ParsedTermsOSA(X), t be Element of TS
  DTConOSA(X) holds t in OSClass(R,t)
proof
  let S be locally_directed OrderSortedSign, X be non-empty ManySortedSet of S
  , R be OSCongruence of ParsedTermsOSA(X), t be Element of TS DTConOSA(X);
  set PTA = ParsedTermsOSA(X);
  reconsider x = t as Element of (the Sorts of PTA).(LeastSort t) by Def12;
  OSClass(R,t) = OSClass(R,x) by Def27
    .= Class( CompClass(R, CComp(LeastSort t)), x);
  hence thesis by EQREL_1:20,OSALG_4:5;
end;
