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 Th42:
  for S be locally_directed regular monotone OrderSortedSign, X
be non-empty ManySortedSet of S, t1,t2 be Element of TS DTConOSA(X) holds t2 in
  OSClass(PTCongruence(X),t1) iff (PTMin X).t2 = (PTMin X).t1
proof
  let S be locally_directed regular monotone OrderSortedSign, X be non-empty
  ManySortedSet of S, t1,t2 be Element of TS DTConOSA(X);
  set R = PTCongruence(X),M = PTMin(X);
  thus t2 in OSClass(R,t1) implies M.t2 = M.t1 by Th41;
  M.t2 in OSClass(R,t2) by Th40;
  then
A1: OSClass(R,t2) = OSClass(R,M.t2) by Th34;
  M.t1 in OSClass(R,t1) by Th40;
  then
A2: OSClass(R,t1) = OSClass(R,M.t1) by Th34;
  assume M.t2 = M.t1;
  hence thesis by A2,A1,Th32;
end;
