theorem
  for j being Integer st j in dom (s.(the_array_sort_of S).M) &
  @M.(@i) in (the generators of G).I & j = @i value_at(C,s) holds
  s.(the_array_sort_of S).M.(@i value_at(C,s)) = s.I.(@M.(@i))
  proof
    let j be Integer;
    assume A1: j in dom (s.(the_array_sort_of S).M);
    assume A2: @M.(@i) in (the generators of G).I;
    assume A3: j = @i value_at(C,s);
    reconsider s1 = s as ManySortedFunction of the generators of G,
    the Sorts of C by AOFA_A00:48;
    consider h being ManySortedFunction of T,C such that
A4: h is_homomorphism T,C & s1 = h||the generators of G by AOFA_A00:def 19;
    s.(the_array_sort_of S).M = @M value_at(C,s) by Th61;
    hence s.(the_array_sort_of S).M.(@i value_at(C,s))
    = (@M value_at(C,s)).(@i value_at(C,s)) by A1,A3,Th74
    .= @M.@i value_at(C,s) by Th79
    .= h.I.(@M.@i) by A4,Th29
    .= ((h.I)|((the generators of G).I)).(@M.@i) by A2,FUNCT_1:49
    .= s.I.(@M.(@i)) by A4,MSAFREE:def 1;
  end;
