reserve a,b,e,r,x,y for Real,
  i,j,k,n,m for Element of NAT,
  x1 for set,
  p,q for FinSequence of REAL,
  A for non empty closed_interval Subset of REAL,
  D,D1,D2 for Division of A,
  f,g for Function of A,REAL,
  T for DivSequence of A;

theorem Th8:
  D <= D1 & i in dom D & j in dom D & i < j implies indx(D1,D,i) <
  indx(D1,D,j)
proof
  assume that
A1: D <= D1 and
A2: i in dom D and
A3: j in dom D and
A4: i < j;
A5: D.i = D1.indx(D1,D,i) by A1,A2,INTEGRA1:def 19;
A6: indx(D1,D,j) in dom D1 by A1,A3,INTEGRA1:def 19;
A7: D.j = D1.indx(D1,D,j) by A1,A3,INTEGRA1:def 19;
A8: indx(D1,D,i) in dom D1 by A1,A2,INTEGRA1:def 19;
  D.i < D.j by A2,A3,A4,SEQM_3:def 1;
  hence thesis by A5,A8,A7,A6,SEQ_4:137;
end;
