reserve i,j,k,n for Nat;
reserve D for non empty set,
  p for Element of D,
  f,g for FinSequence of D;
reserve D for set,
  f for FinSequence of D;

theorem Th3:
  for f being non empty FinSequence holds f/^1 = Del(f,1)
proof
  let f be non empty FinSequence;
  consider i be Nat such that
A1: i+1 = len f by NAT_1:6;
  reconsider i as Element of NAT by ORDINAL1:def 12;
A2: 1 <= len f by A1,NAT_1:11;
  len (f/^1) = len Del(f,1) & for k be Nat st 1 <=k & k <= len (f/^1)
  holds (f/^1).k=Del(f,1).k
  proof
A3: len (f/^1) = (i+1)-1 by A1,A2,RFINSEQ:def 1
      .= i;
    1 in dom f by Th6;
    hence len (f/^1) = len Del(f,1) by A1,A3,FINSEQ_3:109;
A4: 1 in dom f by Th6;
    let k be Nat such that
A5: 1 <=k & k <= len (f/^1);
    k in dom (f/^1) by A5,FINSEQ_3:25;
    hence (f/^1).k= f.(k+1) by A2,RFINSEQ:def 1
      .= Del(f,1).k by A1,A3,A5,A4,FINSEQ_3:111;
  end;
  hence thesis;
end;
