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;
reserve S for non empty non void ManySortedSign,
  A for non-empty finite-yielding MSAlgebra over S,
  V for Variables of A,
  X for SetWithCompoundTerm of S,V;
reserve G1, G2 for Circuit-like non void non empty ManySortedSign,
  f, g for Function,
  C1 for non-empty Circuit of G1,
  C2 for non-empty Circuit of G2;

theorem Th59:
  for G being Circuit-like non void non empty ManySortedSign
  for C being non-empty Circuit of G st C calculates X, A
  for f being SortMap of X, A, C for t being Term of S,V st t in Subtrees X
  for s being State of C holds Following(s, 1+height dom t) is_stable_at f.t &
  for s9 being State of X-Circuit A st s9 = s*f
  for h being CompatibleValuation of s9
  holds Following(s, 1+height dom t).(f.t) = t@(h, A)
proof
  let G be Circuit-like non void non empty ManySortedSign;
  let C be non-empty Circuit of G such that
A1: C calculates X, A;
  let f be SortMap of X, A, C;
  consider g such that
A2: f, g form_embedding_of X-Circuit A, C by A1,Def17;
A3: f preserves_inputs_of X-CircuitStr, G by A1,Def17;
A4: f, g form_morphism_between X-CircuitStr, G by A2;
  let t be Term of S,V such that
A5: t in Subtrees X;
  let s be State of C;
  reconsider s9 = s*f as State of X-Circuit A by A2,Th44;
  reconsider t9 = t as Vertex of X-CircuitStr by A5;
A6: Following(s9, 1+height dom t) is_stable_at t9 by Th21;
A7: Following(s9, 1+height dom t) = Following(s, 1+height dom t)*f by A2,A3
,Th47;
  hence Following(s, 1+height dom t) is_stable_at f.t by A2,A3,A6,Th49;
  let s9 be State of X-Circuit A such that
A8: s9 = s*f;
  let h be CompatibleValuation of s9;
A9: dom f = the carrier of X-CircuitStr by A4;
  Following(s9, 1+height dom t).t9 = t@(h,A) by Th21;
  hence thesis by A7,A8,A9,FUNCT_1:13;
end;
