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 Th36:
  for S be locally_directed OrderSortedSign, X be non-empty
ManySortedSet of S, s be Element of S, t being Element of TS DTConOSA(X), x,x1
being set st x in X.s & t = root-tree [x,s] holds x1 in OSClass(LCongruence(X),
  t) iff x1 = t
proof
  let S be locally_directed OrderSortedSign, X be non-empty ManySortedSet of S
  , s be Element of S, t being Element of TS DTConOSA(X);
  set R = LCongruence(X),R1 = PTCongruence(X);
  let x,x1 being set such that
A1: x in X.s and
A2: t = root-tree [x,s];
  R c= R1 by Def17;
  then OSClass(R,t) c= OSClass(R1,t) by Th35;
  hence x1 in OSClass(R,t) implies x1 = t by A1,A2,Th33;
  assume x1 = t;
  hence thesis by Th32;
end;
