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;
reserve p, q for FinSequence,
  X, Y, x, y for set,
  D for non empty set,
  i, j, k, l, m, n, r for Nat;

theorem
  1<=m & m<=len p implies (m,m)-cut p = <*p.m*>
proof
  assume that
A1: 1<=m and
A2: m<=len p;
  set mp = (m,m)-cut p;
A3: len mp + m = m + 1 by A1,A2,Def1;
  then mp.(0+1) = p.(m+0) by A1,A2,Def1
    .= p.m;
  hence thesis by A3,FINSEQ_1:40;
end;
