let i1 be Element of NAT ; :: thesis: for p being Point of (TOP-REAL 2)
for Y being non empty finite Subset of NAT
for h being non constant standard special_circular_sequence 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 Element of NAT st
( i in dom h & h /. i = (GoB h) * j,(width (GoB h)) ) ) } & i1 = min 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
for h being non constant standard special_circular_sequence 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 Element of NAT st
( i in dom h & h /. i = (GoB h) * j,(width (GoB h)) ) ) } & i1 = min Y holds
((GoB h) * i1,(width (GoB h))) `1 <= p `1
let Y be non empty finite Subset of NAT ; :: thesis: for h being non constant standard special_circular_sequence 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 Element of NAT st
( i in dom h & h /. i = (GoB h) * j,(width (GoB h)) ) ) } & i1 = min Y holds
((GoB h) * i1,(width (GoB h))) `1 <= p `1
let h be non constant standard special_circular_sequence; :: 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 Element of NAT st
( i in dom h & h /. i = (GoB h) * j,(width (GoB h)) ) ) } & i1 = min Y implies ((GoB h) * i1,(width (GoB h))) `1 <= p `1 )
set I = width (GoB h);
A1: 1 <= width (GoB h) by GOBOARD7:35;
assume A2: ( 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 Element of NAT st
( i in dom h & h /. i = (GoB h) * j,(width (GoB h)) ) ) } & i1 = min Y ) ; :: thesis: ((GoB h) * i1,(width (GoB h))) `1 <= p `1
then consider i being Element of NAT such that
A3: 1 <= i and
A4: i + 1 <= len h and
A5: p in LSeg (h /. i),(h /. (i + 1)) by SPPOL_2:14;
A6: p `2 = ((GoB h) * 1,(width (GoB h))) `2 by A2, Th42;
i <= i + 1 by NAT_1:11;
then A7: i <= len h by A4, XXREAL_0:2;
A8: 1 <= i + 1 by A3, XREAL_1:147;

now

per cases ( LSeg h,i is horizontal or LSeg h,i is vertical ) by SPPOL_1:41;

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 A3, A4, TOPREAL1:def 5;
then A9: (h /. i) `2 = (h /. (i + 1)) `2 by SPPOL_1:36;
then A10: p `2 = (h /. i) `2 by A5, GOBOARD7:6;
A11: p `2 = (h /. (i + 1)) `2 by A5, A9, GOBOARD7:6;

now

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

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

then A12: (h /. i) `1 <= p `1 by A5, TOPREAL1:9;
((GoB h) * i1,(width (GoB h))) `1 <= (h /. i) `1 by A2, A6, A1, A3, A7, A10, Th44;
hence ((GoB h) * i1,(width (GoB h))) `1 <= p `1 by A12, XXREAL_0:2; :: thesis: verum

end;

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

then A13: (h /. (i + 1)) `1 <= p `1 by A5, TOPREAL1:9;
((GoB h) * i1,(width (GoB h))) `1 <= (h /. (i + 1)) `1 by A2, A6, A1, A4, A8, A11, Th44;
hence ((GoB h) * i1,(width (GoB h))) `1 <= p `1 by A13, 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 A3, A4, TOPREAL1:def 5;
then A14: (h /. i) `1 = (h /. (i + 1)) `1 by SPPOL_1:37;
then A15: p `1 = (h /. i) `1 by A5, GOBOARD7:5;
A16: 1 <= len (GoB h) by GOBOARD7:34;
A17: p `1 = (h /. (i + 1)) `1 by A5, A14, GOBOARD7:5;

now

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 A18: (h /. i) `2 >= ((GoB h) * 1,(width (GoB h))) `2 by A6, A5, TOPREAL1:10;
(h /. i) `2 <= ((GoB h) * 1,(width (GoB h))) `2 by A3, A7, A16, Th8;
then (h /. i) `2 = ((GoB h) * 1,(width (GoB h))) `2 by A18, XXREAL_0:1;
hence ((GoB h) * i1,(width (GoB h))) `1 <= p `1 by A2, A1, A3, A7, A15, Th44; :: thesis: verum

end;

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

then A19: (h /. (i + 1)) `2 >= ((GoB h) * 1,(width (GoB h))) `2 by A6, A5, TOPREAL1:10;
(h /. (i + 1)) `2 <= ((GoB h) * 1,(width (GoB h))) `2 by A4, A8, A16, Th8;
then (h /. (i + 1)) `2 = ((GoB h) * 1,(width (GoB h))) `2 by A19, XXREAL_0:1;
hence ((GoB h) * i1,(width (GoB h))) `1 <= p `1 by A2, A1, A4, A8, A17, Th44; :: 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