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 Th7:
  (1, len p)-cut p = p
proof
  set cp = (1, len p)-cut p;
  now
A1: 1<=len p +1 by NAT_1:11;
    then
A2: len cp + 1 = len p + 1 by Lm2;
    hence len cp = len p;
    thus len p = len p;
    let i be Nat;
    assume
A3: i in dom cp;
A4: dom cp = Seg len p by A2,FINSEQ_1:def 3;
    then
A5: i<=len p by A3,FINSEQ_1:1;
    0+1<=i by A4,A3,FINSEQ_1:1;
    then ex k being Nat st 0<=k & k<len cp & i=k+1 by A2,A5,Th1;
    hence cp.i = p.i by A1,Lm2;
  end;
  hence thesis by FINSEQ_2:9;
end;
