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));

theorem Th2:
  for I,J being FinSequence-membered set, p,q being FinSequence
  st len p = len q & p <> q holds p^^I misses q^^J
  proof
    let I,J be FinSequence-membered set;
    let p,q be FinSequence;
    assume Z0: len p = len q;
    assume Z1: p <> q;
    assume p^^I meets q^^J;
    then consider a being object such that
A1: a in p^^I & a in q^^J by XBOOLE_0:3;
    consider p1 being Element of I such that
A2: a = p^p1 & p1 in I by A1;
    consider q1 being Element of J such that
A3: a = q^q1 & q1 in J by A1;
    dom p = Seg len p & dom q = Seg len q by FINSEQ_1:def 3;
    then p = (p^p1)|dom p = q by A2,A3,Z0,FINSEQ_1:21;
    hence thesis by Z1;
  end;
