let h be non constant standard special_circular_sequence; :: thesis: for i1 being Nat
for p being Point of (TOP-REAL 2)
for Y being non empty finite Subset of NAT st p `2 = N-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,(width (GoB h))] in Indices (GoB h) & ex i being Nat st
( i in dom h & h /. i = (GoB h) * (j,(width (GoB h))) ) ) } & i1 = max Y holds
((GoB h) * (i1,(width (GoB h)))) `1 >= p `1
let i1 be Nat; :: thesis: for p being Point of (TOP-REAL 2)
for Y being non empty finite Subset of NAT st p `2 = N-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,(width (GoB h))] in Indices (GoB h) & ex i being Nat st
( i in dom h & h /. i = (GoB h) * (j,(width (GoB h))) ) ) } & i1 = max Y holds
((GoB h) * (i1,(width (GoB h)))) `1 >= p `1
let p be Point of (TOP-REAL 2); :: thesis: for Y being non empty finite Subset of NAT st p `2 = N-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,(width (GoB h))] in Indices (GoB h) & ex i being Nat st
( i in dom h & h /. i = (GoB h) * (j,(width (GoB h))) ) ) } & i1 = max Y holds
((GoB h) * (i1,(width (GoB h)))) `1 >= p `1
let Y be non empty finite Subset of NAT; :: thesis: ( p `2 = N-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,(width (GoB h))] in Indices (GoB h) & ex i being Nat st
( i in dom h & h /. i = (GoB h) * (j,(width (GoB h))) ) ) } & i1 = max Y implies ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 )
assume A1: ( p `2 = N-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,(width (GoB h))] in Indices (GoB h) & ex i being Nat st
( i in dom h & h /. i = (GoB h) * (j,(width (GoB h))) ) ) } & i1 = max Y ) ; :: thesis: ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1
then consider i being Nat such that
A2: 1 <= i and
A3: i + 1 <= len h and
A4: p in LSeg ((h /. i),(h /. (i + 1))) by SPPOL_2:14;
A5: p `2 = ((GoB h) * (1,(width (GoB h)))) `2 by A1, Th40;
i <= i + 1 by NAT_1:11;
then A6: i <= len h by A3, XXREAL_0:2;
A7: 1 <= i + 1 by A2, XREAL_1:145;

now :: thesis: ( ( LSeg (h,i) is horizontal & ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 ) or ( LSeg (h,i) is vertical & ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 ) )

per cases ( LSeg (h,i) is horizontal or LSeg (h,i) is vertical ) by SPPOL_1:19;

case LSeg (h,i) is horizontal ; :: thesis: ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1

then LSeg ((h /. i),(h /. (i + 1))) is horizontal by A2, A3, TOPREAL1:def 3;
then A8: (h /. i) `2 = (h /. (i + 1)) `2 by SPPOL_1:15;
then A9: p `2 = (h /. i) `2 by A4, GOBOARD7:6;
A10: p `2 = (h /. (i + 1)) `2 by A4, A8, GOBOARD7:6;

now :: thesis: ( ( (h /. i) `1 >= (h /. (i + 1)) `1 & ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 ) or ( (h /. i) `1 < (h /. (i + 1)) `1 & ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 ) )

per cases ( (h /. i) `1 >= (h /. (i + 1)) `1 or (h /. i) `1 < (h /. (i + 1)) `1 ) ;

case A11: (h /. i) `1 >= (h /. (i + 1)) `1 ; :: thesis: ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1

1 <= width (GoB h) by GOBOARD7:33;
then A12: ((GoB h) * (i1,(width (GoB h)))) `1 >= (h /. i) `1 by A1, A5, A2, A6, A9, Th43;
(h /. i) `1 >= p `1 by A4, A11, TOPREAL1:3;
hence ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 by A12, XXREAL_0:2; :: thesis: verum

end;

case A13: (h /. i) `1 < (h /. (i + 1)) `1 ; :: thesis: ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1

1 <= width (GoB h) by GOBOARD7:33;
then A14: ((GoB h) * (i1,(width (GoB h)))) `1 >= (h /. (i + 1)) `1 by A1, A5, A3, A7, A10, Th43;
(h /. (i + 1)) `1 >= p `1 by A4, A13, TOPREAL1:3;
hence ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 by A14, XXREAL_0:2; :: thesis: verum

end;

end;

end;

hence ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 ; :: thesis: verum

end;

case LSeg (h,i) is vertical ; :: thesis: ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1

then LSeg ((h /. i),(h /. (i + 1))) is vertical by A2, A3, TOPREAL1:def 3;
then A15: (h /. i) `1 = (h /. (i + 1)) `1 by SPPOL_1:16;
then A16: p `1 = (h /. i) `1 by A4, GOBOARD7:5;
A17: 1 <= len (GoB h) by GOBOARD7:32;
A18: p `1 = (h /. (i + 1)) `1 by A4, A15, GOBOARD7:5;

now :: thesis: ( ( (h /. i) `2 >= (h /. (i + 1)) `2 & ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 ) or ( (h /. i) `2 < (h /. (i + 1)) `2 & ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 ) )

per cases ( (h /. i) `2 >= (h /. (i + 1)) `2 or (h /. i) `2 < (h /. (i + 1)) `2 ) ;

case (h /. i) `2 >= (h /. (i + 1)) `2 ; :: thesis: ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1

then A19: (h /. i) `2 >= ((GoB h) * (1,(width (GoB h)))) `2 by A5, A4, TOPREAL1:4;
(h /. i) `2 <= ((GoB h) * (1,(width (GoB h)))) `2 by A2, A6, A17, Th6;
then A20: (h /. i) `2 = ((GoB h) * (1,(width (GoB h)))) `2 by A19, XXREAL_0:1;
1 <= width (GoB h) by GOBOARD7:33;
hence ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 by A1, A2, A6, A16, A20, Th43; :: thesis: verum

end;

case (h /. i) `2 < (h /. (i + 1)) `2 ; :: thesis: ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1

then A21: (h /. (i + 1)) `2 >= ((GoB h) * (1,(width (GoB h)))) `2 by A5, A4, TOPREAL1:4;
(h /. (i + 1)) `2 <= ((GoB h) * (1,(width (GoB h)))) `2 by A3, A7, A17, Th6;
then A22: (h /. (i + 1)) `2 = ((GoB h) * (1,(width (GoB h)))) `2 by A21, XXREAL_0:1;
1 <= width (GoB h) by GOBOARD7:33;
hence ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 by A1, A3, A7, A18, A22, Th43; :: thesis: verum

end;

end;

end;

hence ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 ; :: thesis: verum

end;

end;

end;

hence ((GoB h) * (i1,(width (GoB h)))) `1 >= p `1 ; :: thesis: verum