reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem Th32:
  for p be increasing FinSequence of REAL, n be Element of NAT
  holds n <= len p implies p/^n is increasing FinSequence of REAL
proof
  let p be increasing FinSequence of REAL;
  let n be Element of NAT;
  assume
A1: n <= len p;
  for i,j being Nat
st i in dom (p/^n) & j in dom (p/^n) & i< j holds (p/^n).i < (p
  /^n).j
  proof
    let i,j be Nat;
    assume that
A2: i in dom(p/^n) and
A3: j in dom(p/^n) and
A4: i<j;
A5: i+n<j+n by A4,XREAL_1:6;
A6: j in Seg(len (p/^n)) by A3,FINSEQ_1:def 3;
    then 1 <= j by FINSEQ_1:1;
    then
A7: 1+n <= j+n by XREAL_1:6;
    j <= len (p/^n) by A6,FINSEQ_1:1;
    then j <= len p - n by A1,RFINSEQ:def 1;
    then
A8: j+n <= len p by XREAL_1:19;
    1 <= 1+n by NAT_1:11;
    then 1 <= j+n by A7,XXREAL_0:2;
    then j+n in Seg(len p) by A8,FINSEQ_1:1;
    then
A9: j+n in dom p by FINSEQ_1:def 3;
A10: i in Seg(len (p/^n)) by A2,FINSEQ_1:def 3;
    then 1 <= i by FINSEQ_1:1;
    then
A11: 1+n <= i+n by XREAL_1:6;
    i <= len (p/^n) by A10,FINSEQ_1:1;
    then i <= len p - n by A1,RFINSEQ:def 1;
    then
A12: i+n <= len p by XREAL_1:19;
    1 <= 1+n by NAT_1:11;
    then 1 <= i+n by A11,XXREAL_0:2;
    then i+n in Seg(len p) by A12,FINSEQ_1:1;
    then
A13: i+n in dom p by FINSEQ_1:def 3;
A14: (p/^n).j=p.(j+n) by A1,A3,RFINSEQ:def 1;
    (p/^n).i=p.(i+n) by A1,A2,RFINSEQ:def 1;
    hence thesis by A14,A13,A9,A5,SEQM_3:def 1;
  end;
  hence thesis by SEQM_3:def 1;
end;
