reserve A,B,C,Y,x,y,z for set, U, D for non empty set,
X for non empty Subset of D, d,d1,d2 for Element of D;
reserve P,Q,R for Relation, g for Function, p,q for FinSequence;
reserve f for BinOp of D, i,m,n for Nat;
reserve X for set, f for Function;
reserve U1,U2 for non empty set;
reserve f for BinOp of D;
reserve a,a1,a2,b,b1,b2,A,B,C,X,Y,Z,x,x1,x2,y,y1,y2,z for set,
U,U1,U2,U3 for non empty set, u,u1,u2 for Element of U,
P,Q,R for Relation, f,f1,f2,g,g1,g2 for Function,
k,m,n for Nat, kk,mm,nn for Element of NAT, m1, n1 for non zero Nat,
p, p1, p2 for FinSequence, q, q1, q2 for U-valued FinSequence;

theorem ::#Th44
p^p1^p2 is X-valued implies (p2 is X-valued & p1 is X-valued & p is X-valued)
proof
set q=p^p1^p2; assume q is X-valued; then rng q c= X & rng (p^p1) c= rng q
& rng p2 c= rng q by  FINSEQ_1:29,30; then rng p2 c= X
& rng p c= rng (p^p1) & rng p1 c= rng (p^p1) & rng (p^p1) c= X by
FINSEQ_1:29, 30;
hence thesis by  XBOOLE_1:1;
end;
