reserve
  a,b for object, I,J for set, f for Function, R for Relation,
  i,j,n for Nat, m for (Element of NAT),
  S for non empty non void ManySortedSign,
  s,s1,s2 for SortSymbol of S,
  o for OperSymbol of S,
  X for non-empty ManySortedSet of the carrier of S,
  x,x1,x2 for (Element of X.s), x11 for (Element of X.s1),
  T for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S,
  g for Translation of Free(S,X),s1,s2,
  h for Endomorphism of Free(S,X);
reserve
  r,r1,r2 for (Element of T),
  t,t1,t2 for (Element of Free(S,X));
reserve
  Y for infinite-yielding ManySortedSet of the carrier of S,
  y,y1 for (Element of Y.s), y11 for (Element of Y.s1),
  Q for all_vars_including inheriting_operations free_in_itself
  (Y,S)-terms MSAlgebra over S,
  q,q1 for (Element of Args(o,Free(S,Y))),
  u,u1,u2 for (Element of Q),
  v,v1,v2 for (Element of Free(S,Y)),
  Z for non-trivial ManySortedSet of the carrier of S,
  z,z1 for (Element of Z.s),
  l,l1 for (Element of Free(S,Z)),
  R for all_vars_including inheriting_operations free_in_itself
  (Z,S)-terms MSAlgebra over S,
  k,k1 for Element of Args(o,Free(S,Z));
reserve c,c1,c2 for set, d,d1 for DecoratedTree;

theorem
  for A being disjoint_valued non-empty MSAlgebra over S
  for B being non-empty MSAlgebra over S
  for f1,f2 being ManySortedFunction of A,B st
    for a being Element of A holds f1.a = f2.a
    holds f1 = f2
  proof
    let A be disjoint_valued non-empty MSAlgebra over S;
    let B be non-empty MSAlgebra over S;
    let f1,f2 be ManySortedFunction of A,B;
    assume Z0: for a being Element of A holds f1.a = f2.a;
    let s;
    now
      thus f1.s is Function of (the Sorts of A).s, (the Sorts of B).s &
      f2.s is Function of (the Sorts of A).s, (the Sorts of B).s;
      let a be Element of (the Sorts of A).s;
      thus f1.s.a = f1.a by Th9 .= f2.a by Z0 .= f2.s.a by Th9;
    end;
    hence thesis by FUNCT_2:def 8;
  end;
