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 Th35:
  for S be locally_directed OrderSortedSign, X be non-empty
ManySortedSet of S, R1,R2 be OSCongruence of ParsedTermsOSA(X), t be Element of
  TS DTConOSA(X) holds R1 c= R2 implies OSClass(R1,t) c= OSClass(R2,t)
proof
  let S be locally_directed OrderSortedSign, X be non-empty ManySortedSet of S
  , R1,R2 be OSCongruence of ParsedTermsOSA(X), t be Element of TS DTConOSA(X);
  set s = LeastSort t, PTA = ParsedTermsOSA(X), SPTA = the Sorts of PTA, D =
  DTConOSA(X);
  set CC1 = CComp(s), CR1 = CompClass(R1,CC1);
  reconsider xa=t as Element of SPTA.s by Def12;
  assume
A1: R1 c= R2;
A2: OSClass(R1,t) = OSClass(R1,xa) by Def27
    .= Class( CompClass(R1, CComp(s)), xa);
  let x be object;
  assume x in OSClass(R1,t);
  then [x,xa] in CR1 by A2,EQREL_1:19;
  then consider s1 being Element of S such that
  s1 in CC1 and
A3: [x,xa] in R1.s1 by OSALG_4:def 9;
  reconsider xa = t, xb = x as Element of SPTA.s1 by A3,ZFMISC_1:87;
A4: R1.s1 c= R2.s1 by A1;
  x in SPTA.s1 by A3,ZFMISC_1:87;
  then reconsider t1 = x as Element of TS D by Th15;
  OSClass(R2,t1) = OSClass(R2,xb) by Def27
    .= OSClass(R2,xa) by A3,A4,OSALG_4:12
    .= OSClass(R2,t) by Def27;
  hence thesis by Th34;
end;
