reserve a,x,y for object, A,B for set,
  l,m,n for Nat;

theorem
  for I,J being set, M being ManySortedSet of [:I,J:] for i,j being set
  st i in I & j in J holds (i,j):-> (M.(i,j)) = M|([:{i},{j}:] qua set)
proof
  let I,J be set, M be ManySortedSet of [:I,J:];
  let i,j be set;
  assume i in I & j in J;
  then
A1: [i,j] in [:I,J:] by ZFMISC_1:87;
  thus (i,j):-> (M.(i,j)) = [i,j].--> (M.[i,j])
    .= M|({[i,j]} qua set) by A1,Th7
    .= M|([:{i},{j}:] qua set) by ZFMISC_1:29;
end;
