
theorem Th3:
  for X being non empty set for p being FinSequence of FinTrees X
  for x,t being set st t in rng p holds t <> x-tree p
proof
  let X be non empty set;
  let p be FinSequence of FinTrees X;
  let x,t be set;
  assume
A1: t in rng p;
  then reconsider T = t as Element of FinTrees X;
  reconsider A = dom T as finite Tree;
  defpred P[set] means not contradiction;
  deffunc F(Element of A) = len $1;
  {F(e) where e is Element of A: P[e]} is finite from PRE_CIRC:sch 1;
  then reconsider B = {F(e) where e is Element of A: P[e]} as finite set;
  set e = the Element of A;
A2: F(e) in B;
  B is real-membered
  proof
    let a be object;
    assume a in B;
    then ex e being Element of A st a = F(e);
    hence thesis;
  end;
  then reconsider B as non empty finite real-membered set by A2;
  max B in B by XXREAL_2:def 8;
  then consider e being Element of A such that
A3: max B = len e;
  consider i being object such that
A4: i in dom p and
A5: t = p.i by A1,FUNCT_1:def 3;
  reconsider i as Nat by A4;
  i >= 1 by A4,FINSEQ_3:25;
  then consider j being Nat such that
A6: i = 1+j by NAT_1:10;
  i <= len p by A4,FINSEQ_3:25;
  then
A7: j < len p by A6,NAT_1:13;
A8: <*j*>^e in dom (x-tree p) by A5,A6,A7,TREES_4:11;
  len (<*j*>^e) = 1+len e by FINSEQ_5:8;
  then len (<*j*>^e) > max B by A3,NAT_1:13;
  then len (<*j*>^e) nin B by XXREAL_2:def 8;
  hence thesis by A8;
end;
