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 Th5:
  1 <= i & i <= len h & 1 <= I & I <= width GoB h implies (GoB h)*(
  1,I)`1 <= (h/.i)`1 & (h/.i)`1 <= (GoB h)*(len GoB h,I)`1
proof
  assume that
A1: 1<=i and
A2: i<=len h and
A3: 1 <= I and
A4: I <= width GoB h;
A5: I<=width GoB(Incr(X_axis(h)),Incr(Y_axis(h))) by A4,GOBOARD2:def 2;
  i<=len (X_axis h) by A2,GOBOARD1:def 1;
  then
A6: i in dom (X_axis h) by A1,FINSEQ_3:25;
  then (X_axis(h)).i=(h/.i)`1 by GOBOARD1:def 1;
  then
A7: (h/.i)`1 in rng X_axis h by A6,FUNCT_1:def 3;
A8: GoB h=GoB(Incr(X_axis h),Incr(Y_axis(h))) by GOBOARD2:def 2;
  then 1<=len GoB(Incr(X_axis h),Incr(Y_axis h)) by GOBOARD7:32;
  then
A9: [1,I] in Indices GoB(Incr(X_axis h),Incr(Y_axis h)) by A3,A5,MATRIX_0:30;
A10: 1<=len GoB h by GOBOARD7:32;
  len GoB h <= len GoB(Incr(X_axis h),Incr(Y_axis h)) by GOBOARD2:def 2;
  then
A11: [len GoB h,I] in Indices GoB(Incr(X_axis h),Incr(Y_axis h)) by A3,A5,A10,
MATRIX_0:30;
  (GoB h)*(len GoB h,I)=(GoB(Incr(X_axis h),Incr(Y_axis h)))* (len GoB h,
  I) by GOBOARD2:def 2
    .=|[Incr(X_axis h).len GoB h,Incr(Y_axis(h)).I]| by A11,GOBOARD2:def 1;
  then
A12: (GoB h)*(len GoB h,I)`1 = Incr(X_axis h).len GoB h by EUCLID:52;
  (GoB h)*(1,I)=(GoB(Incr(X_axis(h)),Incr(Y_axis(h))))*(1,I) by GOBOARD2:def 2
    .=|[Incr(X_axis(h)).1,Incr(Y_axis(h)).I]| by A9,GOBOARD2:def 1;
  then
A13: (GoB h)*(1,I)`1=Incr(X_axis(h)).1 by EUCLID:52;
  len GoB h = len Incr (X_axis h) by A8,GOBOARD2:def 1;
  hence thesis by A12,A13,A7,Th4;
end;
