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 Th33:
  for p be increasing FinSequence of REAL, i,j be Element of NAT
holds j in dom p & i <= j implies mid(p,i,j) is increasing FinSequence of REAL
proof
  let p be increasing FinSequence of REAL;
  let i,j be Element of NAT;
  assume that
A1: j in dom p and
A2: i <= j;
  j in Seg(len p) by A1,FINSEQ_1:def 3;
  then j <= len p by FINSEQ_1:1;
  then i <= len p by A2,XXREAL_0:2;
  then p/^(i-'1) is increasing FinSequence of REAL by Th32,NAT_D:44;
  then
A3: (p/^(i-'1))|Seg(j-'i+1) is increasing FinSequence of REAL by FINSEQ_1:18;
  mid(p,i,j)=(p/^(i-'1))|(j-'i+1) by A2,FINSEQ_6:def 3;
  hence thesis by A3,FINSEQ_1:def 16;
end;
