reserve
  a,b for object, I,J for set, f for Function, R for Relation,
  i,j,n for Nat, m for (Element of NAT),
  S for non empty non void ManySortedSign,
  s,s1,s2 for SortSymbol of S,
  o for OperSymbol of S,
  X for non-empty ManySortedSet of the carrier of S,
  x,x1,x2 for (Element of X.s), x11 for (Element of X.s1),
  T for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S,
  g for Translation of Free(S,X),s1,s2,
  h for Endomorphism of Free(S,X);
reserve
  r,r1,r2 for (Element of T),
  t,t1,t2 for (Element of Free(S,X));
reserve
  Y for infinite-yielding ManySortedSet of the carrier of S,
  y,y1 for (Element of Y.s), y11 for (Element of Y.s1),
  Q for all_vars_including inheriting_operations free_in_itself
  (Y,S)-terms MSAlgebra over S,
  q,q1 for (Element of Args(o,Free(S,Y))),
  u,u1,u2 for (Element of Q),
  v,v1,v2 for (Element of Free(S,Y)),
  Z for non-trivial ManySortedSet of the carrier of S,
  z,z1 for (Element of Z.s),
  l,l1 for (Element of Free(S,Z)),
  R for all_vars_including inheriting_operations free_in_itself
  (Z,S)-terms MSAlgebra over S,
  k,k1 for Element of Args(o,Free(S,Z));
reserve c,c1,c2 for set, d,d1 for DecoratedTree;
reserve
  w for (Element of Args(o,T)),
  p,p1 for Element of Args(o,Free(S,X));

theorem Th27:
  for r being SortSymbol of S, y being Element of X.r holds
  x-term is context of y iff r = s & x = y
  proof
    let r be SortSymbol of S, y be Element of X.r;
A0: ([x,s] in {[y,r]} iff [x,s] = [y,r]) & (x-term).{} = [x,s] &
    Coim(x-term, [y,r]) c= dom(x-term) = {{}} & {} in {{}}
    by TARSKI:def 1,TREES_1:29,TREES_4:3,RELAT_1:132;
    (ex a st Coim(x-term, [y,r]) = {a}) implies Coim(x-term, [y,r]) = {{}}
    by A0,ZFMISC_1:33;
    then card Coim(x-term, [y,r]) = 1 implies Coim(x-term, [y,r]) = {{}}
    by CARD_2:42;
    hence thesis by A0,CONTEXT,XTUPLE_0:1,FUNCT_1:def 7;
  end;
