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;

theorem Th53:
  len p = k + 1 & q = p | Seg k implies p = q ^ <* p.(k + 1) *>
proof
  assume that
A1: len p = k + 1 and
A2: q = p | Seg k;
A3: for l being Nat holds l in dom q implies p.l = q.l
   by A2,FUNCT_1:47;
  set r = <* p.(k + 1) *>;
A4: now
    let l be Nat;
    assume l in dom r;
    then l in {1} by FINSEQ_1:2,38;
    then
A5: l = 1 by TARSKI:def 1;
    hence p.(len q + l) = p.(k + 1) by A1,A2,Th51
      .= r.l by A5;
  end;
  len p = len q + 1 by A1,A2,Th51
    .= len q + len<* p.(k + 1) *> by FINSEQ_1:39;
  hence thesis by A3,A4,Th36;
end;
