reserve p, q for Point of TOP-REAL 2,
  r for Real,
  h for non constant standard special_circular_sequence,
  g for FinSequence of TOP-REAL 2,
  f for non empty FinSequence of TOP-REAL 2,
  I, i1, i, j, k for Nat;

theorem Th4:
  for g being FinSequence of REAL st r in rng g holds Incr g.1 <= r
  & r <= Incr g.len Incr g
proof
  let g be FinSequence of REAL;
  assume r in rng g;
  then r in rng Incr g by SEQ_4:def 21;
  then consider x being object such that
A1: x in dom Incr g and
A2: Incr g.x=r by FUNCT_1:def 3;
  reconsider j=x as Nat by A1;
A3: x in Seg len Incr g by A1,FINSEQ_1:def 3;
  then
A4: j <= len Incr g by FINSEQ_1:1;
A5: 1 <= j by A3,FINSEQ_1:1;
  then
A6: 1 <= len Incr g by A4,XXREAL_0:2;
  then 1 in dom Incr g by FINSEQ_3:25;
  hence Incr g.1 <= r by A1,A2,A5,SEQ_4:137;
  len Incr g in dom Incr g by A6,FINSEQ_3:25;
  hence thesis by A1,A2,A4,SEQ_4:137;
end;
