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 Th11:
  rng ((m,n)-cut p) c= rng p
proof
  set c = ((m,n)-cut p);
A1: now
    assume that
A2: 1<=m and
A3: m<=n and
A4: n<=len p;
    n <= n+1 by NAT_1:11;
    then
A5: m <= n+1 by A3,XXREAL_0:2;
    thus thesis
    proof
      let x be object;
      assume x in rng c;
      then consider z being object such that
A6:   z in dom c and
A7:   x = c.z by FUNCT_1:def 3;
      reconsider z as Element of NAT by A6;
A8:   z<=len c by A6,FINSEQ_3:25;
      0+1<=z by A6,FINSEQ_3:25;
      then consider i such that
      0<=i and
A9:   i<len c and
A10:  z=i+1 by A8,Th1;
      m+i<len c +m by A9,XREAL_1:6;
      then m+i<n+1 by A2,A4,A5,Lm2;
      then m+i<=n by NAT_1:13;
      then
A11:  m+i<=len p by A4,XXREAL_0:2;
      1<=m+i by A2,NAT_1:12;
      then
A12:  m+i in dom p by A11,FINSEQ_3:25;
      c.z = p.(m+i) by A2,A4,A5,A9,A10,Lm2;
      hence thesis by A7,A12,FUNCT_1:def 3;
    end;
  end;
  now
    assume not (1<=m & m<=n & n<=len p);
    then c = {} by Def1;
    then rng c = {};
    hence thesis;
  end;
  hence thesis by A1;
end;
