
theorem Th18:
  for I1,I2 being non empty set,
  A being ManySortedSet of I1, B being ManySortedSet of I2,
  o being Element of I2 st B.o <> {}
  for m being Element of B.o, f being Function of I1,I2 st f = I1 --> o holds
  the set of all  [o9,A.o9 --> m] where o9 is Element of I1
  is MSUnTrans of f,A,B
proof
  let I1,I2 be non empty set,
  A be ManySortedSet of I1, B be ManySortedSet of I2,
  o be Element of I2 such that
A1: B.o <> {};
  let m be Element of B.o, f be Function of I1,I2 such that
A2: f = I1 --> o;
  defpred P[set] means not contradiction;
  deffunc F(set) = A.$1 --> m;
  reconsider Xm = { [o9,F(o9)] where o9 is Element of I1:
  P[o9] } as Function from ALTCAT_2:sch 1;
A3: Xm = { [o9,F(o9)] where o9 is Element of I1: P[o9] };
  dom Xm = { o9 where o9 is Element of I1: P[o9] } from ALTCAT_2:sch 2(
  A3)
    .= I1 by DOMAIN_1:18;
  then reconsider Xm as ManySortedSet of I1 by PARTFUN1:def 2,RELAT_1:def 18;
  deffunc F(set) = A.$1 --> m;
A4: Xm = { [o9,F(o9)] where o9 is Element of I1: P[o9] };
  now
    let i be object;
    assume
A5: i in I1;
    then reconsider o9 = i as Element of I1;
A6: P[o9];
A7: i in dom f by A2,A5;
    f.i = o by A2,A5,FUNCOP_1:7;
    then m in B.(f.i) by A1;
    then
A8: m in (B*f).i by A7,FUNCT_1:13;
    Xm.o9 = F(o9) from ALTCAT_2:sch 3(A4,A6);
    hence Xm.i is Function of A.i, (B*f).i by A8,FUNCOP_1:45;
  end;
  then Xm is ManySortedFunction of A,B*f by PBOOLE:def 15;
  hence thesis by Def4;
end;
