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 Th3:
  p in rng h implies ex i be Nat st 1 <= i & i+1 <= len h & h.i = p
proof
A1: 4 < len h by GOBOARD7:34;
A2: 1 < len h by GOBOARD7:34,XXREAL_0:2;
  assume p in rng h;
  then consider x be object such that
A3: x in dom h and
A4: p = h.x by FUNCT_1:def 3;
  reconsider i = x as Nat by A3;
A5: 1 <= i by A3,FINSEQ_3:25;
  set j = S_Drop (i,h);
A6: i <= len h by A3,FINSEQ_3:25;
  1 <= j & j+1 <= len h & h.j = p
  proof
A7: i <= len h -' 1 + 1 by A5,A6,XREAL_1:235,XXREAL_0:2;
    per cases by A7,NAT_1:8;
    suppose
A8:   i <= len h-'1;
      then j = i by A5,JORDAN4:22;
      then j + 1 <= len h -' 1 + 1 by A8,XREAL_1:7;
      hence thesis by A4,A5,A2,A8,JORDAN4:22,XREAL_1:235;
    end;
    suppose
A9:   i = len h-'1+1;
      then i = len h by A1,XREAL_1:235,XXREAL_0:2;
      then i mod (len h -'1) = 1 by A1,Th2,XXREAL_0:2;
      then
A10:  j = 1 by JORDAN4:def 1;
A11:  1 <= len h by GOBOARD7:34,XXREAL_0:2;
      then
A12:  len h in dom h by FINSEQ_3:25;
      1 in dom h by A11,FINSEQ_3:25;
      then h.1 = h/.1 by PARTFUN1:def 6
        .= h/.len h by FINSEQ_6:def 1
        .= h.len h by A12,PARTFUN1:def 6;
      hence thesis by A4,A1,A9,A10,XREAL_1:235,XXREAL_0:2;
    end;
  end;
  hence thesis;
end;
