
theorem Th1:
  for A be non empty set, B be set, D be thin_cylinder of A,B holds
  ex Bo being Subset of B,yo being Function of Bo,A st Bo is finite & D = { y
  where y is Function of B,A : y|Bo = yo }
proof
  let A be non empty set, B be set, D be thin_cylinder of A,B;
  consider Bo being Subset of B,yo being Function of Bo,A such that
A1: Bo is finite and
A2: D = cylinder0(A,B,Bo,yo) by Def2;
  D = { y where y is Function of B,A : y|Bo = yo } by A2,Def1;
  hence thesis by A1;
end;
