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 Th15:
  for I being set
  for A,B being ManySortedSet of I st A c= B
  for f being FinSequence of I holds A*f c= B*f qua ManySortedSet of dom f
  proof
    let I be set;
    let A,B be ManySortedSet of I;
    assume A1: A c= B;
    let f be FinSequence of I;
    let j be object;
    assume A2: j in dom f;
    then (A*f).j = A.(f.j) & (B*f).j = B.(f.j) by FUNCT_1:13;
    hence (A*f).j c= (B*f).j by A1,A2,FUNCT_1:102;
  end;
