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 = S-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,1] in Indices (GoB h) & ex i being Element of NAT st
( i in dom h & h /. i = (GoB h) * j,1 ) ) } & i1 = min Y holds
((GoB h) * i1,1) `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 = S-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,1] in Indices (GoB h) & ex i being Element of NAT st
( i in dom h & h /. i = (GoB h) * j,1 ) ) } & i1 = min Y holds
((GoB h) * i1,1) `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 = S-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,1] in Indices (GoB h) & ex i being Element of NAT st
( i in dom h & h /. i = (GoB h) * j,1 ) ) } & i1 = min Y holds
((GoB h) * i1,1) `1 <= p `1
let h be non constant standard special_circular_sequence; :: thesis: ( p `2 = S-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,1] in Indices (GoB h) & ex i being Element of NAT st
( i in dom h & h /. i = (GoB h) * j,1 ) ) } & i1 = min Y implies ((GoB h) * i1,1) `1 <= p `1 )
assume A1: ( p `2 = S-bound (L~ h) & p in L~ h & Y = { j where j is Element of NAT : ( [j,1] in Indices (GoB h) & ex i being Element of NAT st
( i in dom h & h /. i = (GoB h) * j,1 ) ) } & i1 = min Y ) ; :: thesis: ((GoB h) * i1,1) `1 <= p `1
then A2: p `2 = ((GoB h) * 1,1) `2 by Th40;
consider i being Element of NAT such that
A3: ( 1 <= i & i + 1 <= len h & p in LSeg (h /. i),(h /. (i + 1)) ) by A1, SPPOL_2:14;
i <= i + 1 by NAT_1:11;
then A4: i <= len h by A3, XXREAL_0:2;
A5: ( 1 <= len (GoB h) & 1 <= width (GoB h) ) by GOBOARD7:34, GOBOARD7:35;
A6: 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) * i1,1) `1 <= p `1

then LSeg (h /. i),(h /. (i + 1)) is horizontal by A3, TOPREAL1:def 5;
then (h /. i) `2 = (h /. (i + 1)) `2 by SPPOL_1:36;
then A7: ( p `2 = (h /. (i + 1)) `2 & p `2 = (h /. i) `2 ) by A3, 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) * i1,1) `1 <= p `1

then A8: ( (h /. i) `1 <= p `1 & p `1 <= (h /. (i + 1)) `1 ) by A3, TOPREAL1:9;
((GoB h) * i1,1) `1 <= (h /. i) `1 by A1, A2, A3, A4, A5, A7, Th44;
hence ((GoB h) * i1,1) `1 <= p `1 by A8, XXREAL_0:2; :: thesis: verum

end;

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

then A9: ( (h /. (i + 1)) `1 <= p `1 & p `1 <= (h /. i) `1 ) by A3, TOPREAL1:9;
((GoB h) * i1,1) `1 <= (h /. (i + 1)) `1 by A1, A2, A3, A5, A6, A7, Th44;
hence ((GoB h) * i1,1) `1 <= p `1 by A9, XXREAL_0:2; :: thesis: verum

end;

end;

end;

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

end;

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

then LSeg (h /. i),(h /. (i + 1)) is vertical by A3, TOPREAL1:def 5;
then (h /. i) `1 = (h /. (i + 1)) `1 by SPPOL_1:37;
then A10: ( p `1 = (h /. (i + 1)) `1 & p `1 = (h /. i) `1 ) by A3, 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) * i1,1) `1 <= p `1

then A11: (h /. i) `2 <= ((GoB h) * 1,1) `2 by A2, A3, TOPREAL1:10;
(h /. i) `2 >= ((GoB h) * 1,1) `2 by A3, A4, A5, Th8;
then (h /. i) `2 = ((GoB h) * 1,1) `2 by A11, XXREAL_0:1;
hence ((GoB h) * i1,1) `1 <= p `1 by A1, A3, A4, A5, A10, Th44; :: thesis: verum

end;

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

then A12: (h /. (i + 1)) `2 <= ((GoB h) * 1,1) `2 by A2, A3, TOPREAL1:10;
(h /. (i + 1)) `2 >= ((GoB h) * 1,1) `2 by A3, A5, A6, Th8;
then (h /. (i + 1)) `2 = ((GoB h) * 1,1) `2 by A12, XXREAL_0:1;
hence ((GoB h) * i1,1) `1 <= p `1 by A1, A3, A5, A6, A10, Th44; :: thesis: verum

end;

end;

end;

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

end;

end;

end;

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