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;
reserve D for non empty set;

theorem
  for x being set,f being FinSequence of D st 1<=len f holds
  (f^<*x*>).1=f.1 & (f^<*x*>).1=f/.1 & (<*x*>^f).(len f +1)=f.len f &
  (<*x*>^f).(len f+1)=f/.len f
proof
  let x be set,f be FinSequence of D;
  assume
A1: 1<=len f;
  then
A2: len f in dom f by FINSEQ_3:25;
  1 in dom f by A1,FINSEQ_3:25;
  then
A3: (f^<*x*>).1=f.1 by FINSEQ_1:def 7;
  (<*x*>^f).(len f +1) =(<*x*>^f).(len <*x*>+len f) by FINSEQ_1:39
    .=f.len f by A2,FINSEQ_1:def 7;
  hence thesis by A1,A3,FINSEQ_4:15;
end;
