reserve UA for Universal_Algebra,
  f, g for Function of UA, UA;
reserve I for set,
  A, B, C for ManySortedSet of I;

theorem Th18:
  A is_transformable_to B implies (Funcs)(A,B) is non-empty
proof
  assume
A1: A is_transformable_to B;
A2: for i be set st i in I holds Funcs(A.i, B.i) <> {}
  proof
    let i be set;
    assume i in I;
    then B.i = {} implies A.i = {} by A1;
    hence thesis by FUNCT_2:8;
  end;
  for i be object st i in I holds (Funcs)(A,B).i <> {}
  proof
    let i be object;
    assume
A3: i in I;
    then (Funcs)(A,B).i = Funcs(A.i, B.i) by PBOOLE:def 17;
    hence thesis by A2,A3;
  end;
  then for i be object st i in I holds (Funcs)(A,B).i is non empty;
  hence thesis by PBOOLE:def 13;
end;
