let i1 be 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 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 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 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 Nat st

( i in dom h & h /. i = (GoB h) * (j,1) ) ) } & i1 = min Y implies ((GoB h) * (i1,1)) `1 <= p `1 )

A1: 1 <= width (GoB h) by GOBOARD7:33;

assume A2: ( 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 Nat st

( i in dom h & h /. i = (GoB h) * (j,1) ) ) } & i1 = min Y ) ; :: thesis: ((GoB h) * (i1,1)) `1 <= p `1

then consider i being 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: 1 <= i + 1 by A3, XREAL_1:145;

i <= i + 1 by NAT_1:11;

then A7: i <= len h by A4, XXREAL_0:2;

A8: p `2 = ((GoB h) * (1,1)) `2 by A2, Th38;

A9: 1 <= len (GoB h) by GOBOARD7:32;

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 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 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 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 Nat st

( i in dom h & h /. i = (GoB h) * (j,1) ) ) } & i1 = min Y implies ((GoB h) * (i1,1)) `1 <= p `1 )

A1: 1 <= width (GoB h) by GOBOARD7:33;

assume A2: ( 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 Nat st

( i in dom h & h /. i = (GoB h) * (j,1) ) ) } & i1 = min Y ) ; :: thesis: ((GoB h) * (i1,1)) `1 <= p `1

then consider i being 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: 1 <= i + 1 by A3, XREAL_1:145;

i <= i + 1 by NAT_1:11;

then A7: i <= len h by A4, XXREAL_0:2;

A8: p `2 = ((GoB h) * (1,1)) `2 by A2, Th38;

A9: 1 <= len (GoB h) by GOBOARD7:32;

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

hence
((GoB h) * (i1,1)) `1 <= p `1
; :: thesis: verumper cases
( LSeg (h,i) is horizontal or LSeg (h,i) is vertical )
by SPPOL_1:19;

end;

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, A4, TOPREAL1:def 3;

then A10: (h /. i) `2 = (h /. (i + 1)) `2 by SPPOL_1:15;

then A11: p `2 = (h /. i) `2 by A5, GOBOARD7:6;

A12: p `2 = (h /. (i + 1)) `2 by A5, A10, GOBOARD7:6;

end;then A10: (h /. i) `2 = (h /. (i + 1)) `2 by SPPOL_1:15;

then A11: p `2 = (h /. i) `2 by A5, GOBOARD7:6;

A12: p `2 = (h /. (i + 1)) `2 by A5, A10, GOBOARD7:6;

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

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

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, A4, TOPREAL1:def 3;

then A15: (h /. i) `1 = (h /. (i + 1)) `1 by SPPOL_1:16;

then A16: p `1 = (h /. i) `1 by A5, GOBOARD7:5;

A17: p `1 = (h /. (i + 1)) `1 by A5, A15, GOBOARD7:5;

end;then A15: (h /. i) `1 = (h /. (i + 1)) `1 by SPPOL_1:16;

then A16: p `1 = (h /. i) `1 by A5, GOBOARD7:5;

A17: p `1 = (h /. (i + 1)) `1 by A5, A15, GOBOARD7:5;

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

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

end;

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

then A18:
(h /. i) `2 <= ((GoB h) * (1,1)) `2
by A8, A5, TOPREAL1:4;

(h /. i) `2 >= ((GoB h) * (1,1)) `2 by A3, A7, A9, Th6;

then (h /. i) `2 = ((GoB h) * (1,1)) `2 by A18, XXREAL_0:1;

hence ((GoB h) * (i1,1)) `1 <= p `1 by A2, A3, A7, A1, A16, Th42; :: thesis: verum

end;(h /. i) `2 >= ((GoB h) * (1,1)) `2 by A3, A7, A9, Th6;

then (h /. i) `2 = ((GoB h) * (1,1)) `2 by A18, XXREAL_0:1;

hence ((GoB h) * (i1,1)) `1 <= p `1 by A2, A3, A7, A1, A16, Th42; :: thesis: verum

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

then A19:
(h /. (i + 1)) `2 <= ((GoB h) * (1,1)) `2
by A8, A5, TOPREAL1:4;

(h /. (i + 1)) `2 >= ((GoB h) * (1,1)) `2 by A4, A9, A6, Th6;

then (h /. (i + 1)) `2 = ((GoB h) * (1,1)) `2 by A19, XXREAL_0:1;

hence ((GoB h) * (i1,1)) `1 <= p `1 by A2, A4, A1, A6, A17, Th42; :: thesis: verum

end;(h /. (i + 1)) `2 >= ((GoB h) * (1,1)) `2 by A4, A9, A6, Th6;

then (h /. (i + 1)) `2 = ((GoB h) * (1,1)) `2 by A19, XXREAL_0:1;

hence ((GoB h) * (i1,1)) `1 <= p `1 by A2, A4, A1, A6, A17, Th42; :: thesis: verum