let j be Element of NAT ; :: thesis: for p being Point of (TOP-REAL 2)
for G being Go-board st j < width G & p in Int (h_strip G,j) holds
p `2 < (G * 1,(j + 1)) `2
let p be Point of (TOP-REAL 2); :: thesis: for G being Go-board st j < width G & p in Int (h_strip G,j) holds
p `2 < (G * 1,(j + 1)) `2
let G be Go-board; :: thesis: ( j < width G & p in Int (h_strip G,j) implies p `2 < (G * 1,(j + 1)) `2 )
assume that
A1: j < width G and
A2: p in Int (h_strip G,j) ; :: thesis: p `2 < (G * 1,(j + 1)) `2

per cases ( j = 0 or j >= 1 ) by NAT_1:14;

suppose j = 0 ; :: thesis: p `2 < (G * 1,(j + 1)) `2

then Int (h_strip G,j) = { |[r,s]| where r, s is Real : s < (G * 1,(j + 1)) `2 } by Th18;
then consider r, s being Real such that
A3: p = |[r,s]| and
A4: (G * 1,(j + 1)) `2 > s by A2;
thus p `2 < (G * 1,(j + 1)) `2 by A3, A4, EUCLID:56; :: thesis: verum

end;

suppose j >= 1 ; :: thesis: p `2 < (G * 1,(j + 1)) `2

then Int (h_strip G,j) = { |[r,s]| where r, s is Real : ( (G * 1,j) `2 < s & s < (G * 1,(j + 1)) `2 ) } by A1, Th20;
then consider r, s being Real such that
A5: p = |[r,s]| and
(G * 1,j) `2 < s and
A6: s < (G * 1,(j + 1)) `2 by A2;
thus p `2 < (G * 1,(j + 1)) `2 by A5, A6, EUCLID:56; :: thesis: verum

end;

end;