let p be Point of (TOP-REAL 2); :: thesis: for h being non constant standard special_circular_sequence
for I being Element of NAT st p in L~ h & 1 <= I & I <= width (GoB h) holds
((GoB h) * 1,I) `1 <= p `1
let h be non constant standard special_circular_sequence; :: thesis: for I being Element of NAT st p in L~ h & 1 <= I & I <= width (GoB h) holds
((GoB h) * 1,I) `1 <= p `1
let I be Element of NAT ; :: thesis: ( p in L~ h & 1 <= I & I <= width (GoB h) implies ((GoB h) * 1,I) `1 <= p `1 )
assume A1: ( p in L~ h & 1 <= I & I <= width (GoB h) ) ; :: thesis: ((GoB h) * 1,I) `1 <= p `1
then consider i being Element of NAT such that
A2: ( 1 <= i & i + 1 <= len h & p in LSeg (h /. i),(h /. (i + 1)) ) by SPPOL_2:14;
i <= i + 1 by NAT_1:11;
then i <= len h by A2, XXREAL_0:2;
then A3: ((GoB h) * 1,I) `1 <= (h /. i) `1 by A1, A2, Th7;
1 <= i + 1 by NAT_1:11;
then A4: ((GoB h) * 1,I) `1 <= (h /. (i + 1)) `1 by A1, A2, Th7;

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) * 1,I) `1 <= p `1

then (h /. i) `1 <= p `1 by A2, TOPREAL1:9;
hence ((GoB h) * 1,I) `1 <= p `1 by A3, XXREAL_0:2; :: thesis: verum

end;

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

then (h /. (i + 1)) `1 <= p `1 by A2, TOPREAL1:9;
hence ((GoB h) * 1,I) `1 <= p `1 by A4, XXREAL_0:2; :: thesis: verum

end;

end;

end;

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