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 Th40: f is X*-valued implies f.x in X*
proof
assume f is X*-valued; then
A1: rng f c= X*;
per cases;
suppose A2: x in dom f; then reconsider D=dom f as non empty set;
reconsider e=x as Element of D by A2;
reconsider ff=f as Function of D, X* by FUNCT_2:2, A1;
ff.e is Element of X*; hence thesis;
end;
suppose not x in dom f; then f.x = {} by FUNCT_1:def 2;
hence thesis by FINSEQ_1:49;
end;
end;
