reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;

theorem
  for I being non empty set, J being set
  for A,B being ManySortedSet of I st A c= B
  for f being Function of J,I holds A*f c= B*f qua ManySortedSet of J
  proof
    let I be non empty set, J be set;
    let A,B be ManySortedSet of I;
    assume A1: A c= B;
    let f be Function of J,I;
    let j be object;
    assume A2: j in J;
    then (A*f).j = A.(f.j) & (B*f).j = B.(f.j) by FUNCT_2:15;
    hence (A*f).j c= (B*f).j by A1,A2,FUNCT_2:5;
  end;
