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 q = {[i,x]} holds Seq q = <*x*>
proof
  let q be FinSubsequence;
  assume
A1: q = {[i,x]};
  then [i,x] in q by TARSKI:def 1;
  then
A2: i in dom q by XTUPLE_0:def 12;
  ex k be Nat st dom q c= Seg k by FINSEQ_1:def 12;
  then i >= 0 qua Nat+1 by A2,FINSEQ_1:1;
  then
A3: i > 0;
  then reconsider p = {[i,x]} as FinSubsequence by FINSEQ_1:96;
A4: Seq q = q* Sgm dom q by FINSEQ_1:def 15;
  dom p = {i} by RELAT_1:9;
  then Seq p = {[i,x]}*<*i*> by A1,A3,A4,Th42
    .= <*{[i,x]}.i*> by A1,A2,FINSEQ_2:34
    .= <*x*> by GRFUNC_1:6;
  hence thesis by A1;
end;
