reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;
reserve
  v,v1,v2 for FinSequence of REAL,
  n,m,k for Nat,
  x for set;

theorem
  for n being Nat for f being FinSequence holds 1 <= n & n+2 <= len f
  iff n in dom f & n+1 in dom f & n+2 in dom f
proof
  let n be Nat;
  let f be FinSequence;
  thus 1<=n & n+2 <= len f implies n in dom f & n+1 in dom f & n+2 in dom f
  proof
    assume that
A1: 1<=n and
A2: n+2 <= len f;
    n+1<=n+1+1 by NAT_1:11;
    then
A3: 1<=n+1 & n+1<=len f by A2,NAT_1:11,XXREAL_0:2;
    n<=n+2 by NAT_1:11;
    then 1<=n+2 & n<=len f by A1,A2,XXREAL_0:2;
    hence thesis by A1,A2,A3,FINSEQ_3:25;
  end;
  assume that
A4: n in dom f and
  n+1 in dom f and
A5: n+2 in dom f;
  thus thesis by A4,A5,FINSEQ_3:25;
end;
