reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;

theorem
  f1^f2 just_once_values p & p in rng f1 implies f1 just_once_values p
proof
  assume that
A1: f1^f2 just_once_values p and
A2: p in rng f1;
  (f1^f2)|--p = (f1|--p)^f2 by A2,Th8;
  then
A3: rng((f1^f2)|--p) = rng(f1|--p) \/ rng f2 by FINSEQ_1:31;
  not p in rng((f1^f2)|--p) by A1,FINSEQ_4:45;
  then not p in rng(f1|--p) by A3,XBOOLE_0:def 3;
  hence thesis by A2,FINSEQ_4:45;
end;
