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 Th34:
  for S be locally_directed OrderSortedSign, X be non-empty
ManySortedSet of S, R be OSCongruence of ParsedTermsOSA(X), t1,t2 be Element of
  TS DTConOSA(X) holds t2 in OSClass(R,t1) iff OSClass(R,t1) = OSClass(R,t2)
proof
  let S be locally_directed OrderSortedSign, X be non-empty ManySortedSet of S
  , R be OSCongruence of ParsedTermsOSA(X), t1,t2 be Element of TS DTConOSA(X);
  set PTA = ParsedTermsOSA(X), SPTA = (the Sorts of PTA);
  reconsider x1=t1 as Element of SPTA.(LeastSort t1) by Def12;
  set CC1 = CComp(LeastSort t1), CR1 = CompClass(R,CC1);
A1: OSClass(R,t1) = OSClass(R,x1) by Def27
    .= Class( CompClass(R, CComp(LeastSort t1)), x1);
  hereby
    assume t2 in OSClass(R,t1);
    then [t2,x1] in CR1 by A1,EQREL_1:19;
    then consider s2 being Element of S such that
    s2 in CC1 and
A2: [t2,x1] in R.s2 by OSALG_4:def 9;
    reconsider x11=x1,x22=t2 as Element of SPTA.s2 by A2,ZFMISC_1:87;
    thus OSClass(R,t1) = OSClass(R,x11) by Def27
      .= OSClass(R,x22) by A2,OSALG_4:12
      .= OSClass(R,t2) by Def27;
  end;
  assume OSClass(R,t1) = OSClass(R,t2);
  hence thesis by Th32;
end;
