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

theorem Th19:
  A is_transformable_to B implies
  for x be set st x in MSFuncs(A,B) holds
  x is ManySortedFunction of A,B
proof
  assume
A1: A is_transformable_to B;
  set f = (Funcs)(A,B);
  let x be set;
  assume x in MSFuncs(A,B);
  then x in product f by A1,Def4;
  then consider g be Function such that
A2: x = g and
A3: dom g = dom f and
A4: for i be object st i in dom f holds g.i in f.i by CARD_3:def 5;
A5: dom f = I by PARTFUN1:def 2;
A6: for i be set st i in I holds g.i in Funcs(A.i, B.i)
  proof
    let i be set;
    assume
A7: i in I;
    then (Funcs)(A,B).i = Funcs(A.i,B.i) by PBOOLE:def 17;
    hence thesis by A4,A5,A7;
  end;
A8: for i be set st i in I holds ex F be Function st F = g.i & dom F = A.i
  & rng F c= B.i
  proof
    let i be set;
    assume i in I;
    then g.i in Funcs(A.i, B.i) by A6;
    hence thesis by FUNCT_2:def 2;
  end;
A9: for i be object st i in I holds g.i is Function of A.i, B.i
  proof
    let i be object;
    assume
A10: i in I;
    ex F be Function st F = g.i & dom F = A.i & rng F c= B.i by A8,A10;
    hence thesis by FUNCT_2:2;
  end;
  dom g = I by A3,PARTFUN1:def 2;
  then g is ManySortedSet of I by PARTFUN1:def 2,RELAT_1:def 18;
  hence thesis by A2,A9,PBOOLE:def 15;
end;
