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

per cases ( j = width G or j < width G ) by A1, XXREAL_0:1;

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

then Int (h_strip G,j) = { |[r,s]| where r, s is Real : (G * 1,j) `2 < s } by Th19;
then consider r, s being Real such that
A3: p = |[r,s]| and
A4: (G * 1,j) `2 < s by A2;
thus p `2 > (G * 1,j) `2 by A3, A4, EUCLID:56; :: thesis: verum

end;

suppose j < width G ; :: thesis: p `2 > (G * 1,j) `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
A6: (G * 1,j) `2 < s and
s < (G * 1,(j + 1)) `2 by A2;
thus p `2 > (G * 1,j) `2 by A5, A6, EUCLID:56; :: thesis: verum

end;

end;