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;
reserve J for Nat;
reserve n for Nat;
reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;
reserve n for Nat;

theorem
  for p being FinSequence, k being Nat st k in dom p
  for i being Nat st 1 <= i & i <= k holds i in dom p
proof
  let p be FinSequence, k be Nat;
  assume
A1: k in dom p;
  let i be Nat;
  assume that
A2: 1 <= i and
A3: i <= k;
  k <= len p by A1,Th25;
  then i <= len p by A3,XXREAL_0:2;
  hence thesis by A2,Th25;
end;
