reserve G for Graph,
  k, m, n for Nat;

theorem
  for p,q being FinSequence st n<=len p holds (1,n)-cut p = (1,n)-cut (p ^q)
proof
  let p,q be FinSequence;
  assume
A1: n<=len p;
  per cases;
  suppose
A2: n < 1;
    then (1,n)-cut p = {} by FINSEQ_6:def 4;
    hence thesis by A2,FINSEQ_6:def 4;
  end;
  suppose
A3: 1<=n;
    set cp = (1,n)-cut p, cpq = (1,n)-cut (p^q);
    now
A4:   len cp +1 = n+1 by A1,A3,FINSEQ_6:def 4;
      len (p^q) = len p + len q by FINSEQ_1:22;
      then
A5:   n<=len (p^q) by A1,NAT_1:12;
      then
A6:   len cpq +1 = n +1 by A3,FINSEQ_6:def 4;
      hence len cp = len cpq by A4;
      let k be Nat;
      assume that
A7:   1 <=k and
A8:   k <= len cp;
      k<=len p by A1,A4,A8,XXREAL_0:2;
      then
A9:   k in dom p by A7,FINSEQ_3:25;
      0+1 = 1;
      then
A10:  ex i being Nat st 0<=i & i<len cp & k=i+1 by A7,A8,FINSEQ_6:127;
      hence cp.k = p.k by A1,A3,FINSEQ_6:def 4
        .= (p^q).k by A9,FINSEQ_1:def 7
        .= cpq.k by A3,A5,A4,A6,A10,FINSEQ_6:def 4;
    end;
    hence thesis by FINSEQ_1:14;
  end;
end;
