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 Th6:
  1 <= i & i <= len h & 1 <= I & I <= len GoB h implies (GoB h)*(I,
  1)`2 <= (h/.i)`2 & (h/.i)`2 <= (GoB h)*(I,width GoB h)`2
proof
  assume that
A1: 1<=i and
A2: i<=len h and
A3: 1 <= I and
A4: I <= len GoB h;
A5: GoB h=GoB(Incr(X_axis h),Incr(Y_axis h)) by GOBOARD2:def 2;
  then
A6: 1<=width GoB(Incr(X_axis h),Incr(Y_axis h)) by GOBOARD7:33;
  i <= len Y_axis h by A2,GOBOARD1:def 2;
  then
A7: i in dom Y_axis h by A1,FINSEQ_3:25;
  then (Y_axis h).i = (h/.i)`2 by GOBOARD1:def 2;
  then
A8: (h/.i)`2 in rng Y_axis h by A7,FUNCT_1:def 3;
  1<=width GoB h by GOBOARD7:33;
  then
A9: [I,width GoB h] in Indices GoB(Incr(X_axis h),Incr(Y_axis h)) by A3,A4,A5,
MATRIX_0:30;
  (GoB h)*(I,width GoB h)=(GoB(Incr(X_axis h),Incr(Y_axis h)))* (I,width
  GoB h) by GOBOARD2:def 2
    .=|[Incr(X_axis(h)).I,Incr(Y_axis(h)).width GoB h]| by A9,GOBOARD2:def 1;
  then
A10: (GoB h)*(I,width GoB h)`2 = Incr(Y_axis h).width GoB h by EUCLID:52;
  I<=len GoB(Incr(X_axis h),Incr(Y_axis h)) by A4,GOBOARD2:def 2;
  then
A11: [I,1] in Indices GoB(Incr(X_axis h),Incr(Y_axis h)) by A3,A6,MATRIX_0:30;
  (GoB h)*(I,1)=(GoB(Incr(X_axis h),Incr(Y_axis h)))*(I,1) by GOBOARD2:def 2
    .= |[Incr(X_axis(h)).I,Incr(Y_axis(h)).1]| by A11,GOBOARD2:def 1;
  then
A12: (GoB h)*(I,1)`2 = Incr(Y_axis h).1 by EUCLID:52;
  width GoB h = len Incr Y_axis h by A5,GOBOARD2:def 1;
  hence thesis by A10,A12,A8,Th4;
end;
