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

theorem Th16:
  f1^f2 just_once_values p implies p in rng f1 \+\ rng f2
proof
A1: rng(f1^f2) = rng f1 \/ rng f2 by FINSEQ_1:31;
  assume
A2: f1^f2 just_once_values p;
A3: now
    assume
A4: p in rng f1 /\ rng f2;
    then p in rng f1 by XBOOLE_0:def 4;
    then (f1^f2)|--p = (f1|--p)^f2 by Th8;
    then
A5: not p in rng((f1|--p)^f2) by A2,FINSEQ_4:45;
    rng((f1|--p)^f2) = rng(f1|--p) \/ rng f2 by FINSEQ_1:31;
    then not p in rng f2 by A5,XBOOLE_0:def 3;
    hence contradiction by A4,XBOOLE_0:def 4;
  end;
  p in rng(f1^f2) by A2,FINSEQ_4:5;
  then p in (rng f1 \/ rng f2) \ rng f1 /\ rng f2 by A1,A3,XBOOLE_0:def 5;
  hence thesis by XBOOLE_1:101;
end;
