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 Th6:
  for p,q st rng p = rng q & p is increasing & q is increasing
  holds p = q
proof
  let p,q;
  assume
A1: rng p = rng q;
  assume that
A2: p is increasing and
A3: q is increasing;
A4: q is one-to-one by A3;
  p is one-to-one by A2;
  then len p = len q by A1,A4,FINSEQ_1:48;
  hence thesis by A1,A2,A3,SEQ_4:141;
end;
