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));
reserve C for (context of x), C1 for (context of y), C9 for (context of z),
  C11 for (context of x11), C12 for (context of y11), D for context of s,X;
reserve
  S9 for sufficiently_rich non empty non void ManySortedSign,
  s9 for SortSymbol of S9,
  o9 for s9-dependent OperSymbol of S9,
  X9 for non-trivial ManySortedSet of the carrier of S9,
  x9 for (Element of X9.s9);
reserve h1 for x-constant Homomorphism of Free(S,X), T,
  h2 for y-constant Homomorphism of Free(S,Y), Q;

theorem Th77:
  Q is struct-invariant implies
  for p being Element of Args(o,Q) st
  for t being Element of Q st t in rng p holds t is y-omitting
  for t being Element of Q st t = Den(o,Q).p holds t is y-omitting
  proof assume ZZ: Q is struct-invariant;
    let p be Element of Args(o,Q);
    assume Z0: for t being Element of Q st t in rng p holds
    t is y-omitting;
    let t be Element of Q;
    assume t = Den(o,Q).p;
    then
A1: t = (canonical_homomorphism Q).(the_result_sort_of o).(Den(o,Free(S,Y)).p)
    by MSAFREE4:67;
    Args(o,Q) c= Args(o, Free(S,Y)) by MSAFREE4:41;
    then reconsider q = p as Element of Args(o,Free(S,Y));
A2: Den(o,Free(S,Y)).q = o-term q by MSAFREE4:13;
    now let v; assume
A5:   v in rng p;
      then reconsider d = v as Element of Q by RELAT_1:167;
      d is y-omitting by Z0,A5;
      hence v is y-omitting;
    end;
    then
A3: o-term q is y-omitting by Th54A;
    the_sort_of (o-term q) = the_result_sort_of o by Th8;
    then t = (canonical_homomorphism Q).(o-term q) by A1,A2,ABBR;
    hence thesis by ZZ,A3,Th68;
  end;
