reserve S for non empty non void ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S,
  A for non-empty MSAlgebra over S,
  X for non empty Subset of S-Terms V,
  t for Element of X;

theorem Th17:
  for A being finite-yielding non-empty MSAlgebra over S,
  X being non empty Subset of S-Terms V holds X-Circuit A is finite-yielding
proof
  let A be finite-yielding non-empty MSAlgebra over S;
  let t be non empty Subset of S-Terms V;
  let i be object;
  assume i in the carrier of t-CircuitStr;
  then reconsider i as Vertex of t-CircuitStr;
  reconsider u = i as Term of S,V by Th4;
A1: the Sorts of A is finite-yielding by MSAFREE2:def 11;
  (the Sorts of t-Circuit A).i = the_sort_of (i, A) by Def4
    .= (the Sorts of A).the_sort_of u by Def2;
  hence thesis by A1;
end;
