reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;
reserve m,n,k for Nat, R for Relation;
reserve i,j for Nat;
reserve F for Function,
  e,x,y,z for object;
reserve a,b,c for set;
reserve A,B,I for set, X,Y for ManySortedSet of I;

theorem
 for I being non empty set, X being ManySortedSet of I,
     l1,l2 being Element of I, i1,i2 being set
 holds X +*((l1,l2) --> (i1, i2)) = X +* (l1,i1) +* (l2,i2)
proof
 let I be non empty set, X be ManySortedSet of I,
     l1,l2 be Element of I, i1,i2 be set;
A1: dom X = I by PARTFUN1:def 2;
   then
A2: l1 in dom X;
   dom(X +* (l1,i1)) = I by A1,Th29;
   then
A3: l2 in dom(X +* (l1,i1));
  thus X +*((l1,l2) --> (i1, i2)) =X +* ((l1 .--> i1) +* (l2 .--> i2))
    .=X +* (l1 .--> i1) +* (l2 .--> i2) by FUNCT_4:14
    .=X +* (l1,i1) +* (l2 .--> i2) by A2,Def2
    .=X +* (l1,i1) +* (l2,i2) by A3,Def2;
end;
