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

theorem
  for f being trivial FinSequence of D holds
  f is empty or ex x being Element of D st f = <*x*>
proof
  let f be trivial FinSequence of D;
A1: rng f c= D by FINSEQ_1:def 4;
  assume f is non empty;
  then consider x being object such that
A2: f = {x} by ZFMISC_1:131;
A3: 1 in dom f by A2,FINSEQ_5:6;
A4: x in {x} by TARSKI:def 1;
  then consider y, z being object such that
A5: x = [y,z] by A2,RELAT_1:def 1;
  reconsider z as set by TARSKI:1;
  take z;
  z in rng f by A2,A4,A5,XTUPLE_0:def 13;
  hence z is Element of D by A1;
  dom f = {y} by A2,A5,RELAT_1:9;
  then 1 = y by A3,TARSKI:def 1;
  hence f = <*z*> by A2,A5;
end;
