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

theorem Th20:
  A is_transformable_to B implies
  for g be ManySortedFunction of A, B holds g in MSFuncs(A,B)
proof
  assume
A1: A is_transformable_to B;
  set f = (Funcs)(A,B);
  let g be ManySortedFunction of A, B;
A2: dom f = I by PARTFUN1:def 2;
A3: now
    let x be object;
    assume
A4: x in dom f;
    then reconsider i = x as Element of I by PARTFUN1:def 2;
A5: g.i is Function of A.i, B.i by A2,A4,PBOOLE:def 15;
    B.i = {} implies A.i = {} by A1,A2,A4;
    then g.i in Funcs(A.i, B.i) by A5,FUNCT_2:8;
    hence g.x in f.x by A2,A4,PBOOLE:def 17;
  end;
  dom g = I by PARTFUN1:def 2;
  then g in product f by A2,A3,CARD_3:9;
  hence thesis by A1,Def4;
end;
