reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;
reserve n for Nat;
reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;
reserve n for Nat;

theorem
  for q being FinSubsequence st Seq q = <*x*>
     ex i being Element of NAT st q = {[i,x]}
proof
  let q be FinSubsequence;
  assume Seq q = <*x*>;
  then
A1: Seq q = {[1,x]} by FINSEQ_1:def 5;
  then
A2: dom Seq q = {1} by RELAT_1:9;
A3: rng Seq q = {x} by A1,RELAT_1:9;
A4: Seq q = q* Sgm(dom q) by FINSEQ_1:def 15;
A5: rng Sgm(dom q) = dom q by FINSEQ_1:50;
  then
A6: {1} = dom Sgm(dom q) by A2,A4,RELAT_1:27;
A7: rng Seq q = rng q by A4,A5,RELAT_1:28;
  consider n being Nat such that
A8: dom q c= Seg n by FINSEQ_1:def 12;
  Seg card dom q = {1} by A6,Th38;
  then card dom q = card {1} by FINSEQ_1:57;
  then consider y be object such that
A9: dom q = {y} by CARD_1:29;
  y in dom q by A9,TARSKI:def 1;
  then y in Seg n by A8;
  then reconsider y as Element of NAT;
  q = {[y,x]} by A3,A7,A9,RELAT_1:189;
  hence thesis;
end;
