reserve
  S for (4,1) integer bool-correct non empty non void BoolSignature,
  X for non-empty ManySortedSet of the carrier of S,
  T for vf-free integer all_vars_including inheriting_operations free_in_itself
  (X,S)-terms VarMSAlgebra over S,
  C for (4,1) integer bool-correct non-empty image of T,
  G for basic GeneratorSystem over S,X,T,
  A for IfWhileAlgebra of the generators of G,
  I for integer SortSymbol of S,
  x,y,z,m for pure (Element of (the generators of G).I),
  b for pure (Element of (the generators of G).the bool-sort of S),
  t,t1,t2 for Element of T,I,
  P for Algorithm of A,
  s,s1,s2 for Element of C-States(the generators of G);
reserve
  f for ExecutionFunction of A, C-States(the generators of G),
  (\falseC)-States(the generators of G, b);
reserve u for ManySortedFunction of FreeGen T, the Sorts of C;
reserve
  S for 1-1-connectives (4,1) integer (11,1,1)-array 11 array-correct
  bool-correct non empty non void BoolSignature,
  X for non-empty ManySortedSet of the carrier of S,
  T for vf-free all_vars_including inheriting_operations free_in_itself
  (X,S)-terms integer-array non-empty VarMSAlgebra over S,
  C for (11,1,1)-array (4,1) integer bool-correct non-empty image of T,
  G for basic GeneratorSystem over S,X,T,
  A for IfWhileAlgebra of the generators of G,
  I for integer SortSymbol of S,
  x,y,m,i for pure (Element of (the generators of G).I),
  M,N for pure (Element of (the generators of G).the_array_sort_of S),
  b for pure (Element of (the generators of G).the bool-sort of S),
  s,s1 for (Element of C-States(the generators of G));
reserve u for ManySortedFunction of FreeGen T, the Sorts of C;

theorem Th88:
  for n being Nat holds ^(n+1,T,I) = ^(n,T,I)+\1(T,I) &
  ^(-(n+1),T,I) = - ^(n+1,T,I)
  proof
    let n be Nat;
    consider f being Function of INT, (the Sorts of T).I such that
A1: ^(n+1,T,I) = f.(n+1) & f.0 = \0(T,I) &
    for j being Nat, t being Element of T,I st f.j = t
    holds f.(j+1) = t+\1(T,I) & f.(-(j+1)) = -(t+\1(T,I)) by Def15;
    consider g being Function of INT, (the Sorts of T).I such that
A2: ^(n,T,I) = g.n & g.0 = \0(T,I) &
    for j being Nat, t being Element of T,I st g.j = t
    holds g.(j+1) = t+\1(T,I) & g.(-(j+1)) = -(t+\1(T,I)) by Def15;
    consider h being Function of INT, (the Sorts of T).I such that
A3: ^(-(n+1),T,I) = h.(-(n+1)) & h.0 = \0(T,I) &
    for j being Nat, t being Element of T,I st h.j = t
    holds h.(j+1) = t+\1(T,I) & h.(-(j+1)) = -(t+\1(T,I)) by Def15;
A4: f = g by A1,A2,Lm1;
    ^(n,T,I) = f.n by A1,A2,Lm1;
    hence
A5: ^(n+1,T,I) = ^(n,T,I)+\1(T,I) by A1;
    f = h by A1,A3,Lm1;
    hence ^(-(n+1),T,I) = - ^(n+1,T,I) by A3,A5,A4,A2;
  end;
