let j be 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 and
A2: j <= width G and
A3: p in Int (h_strip (G,j)) ; :: thesis: p `2 > (G * (1,j)) `2

per cases ( j = width G or j < width G ) by A2, 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 Th16;
then ex r, s being Real st
( p = |[r,s]| & (G * (1,j)) `2 < s ) by A3;
hence p `2 > (G * (1,j)) `2 by EUCLID:52; :: 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, Th17;
then ex r, s being Real st
( p = |[r,s]| & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) by A3;
hence p `2 > (G * (1,j)) `2 by EUCLID:52; :: thesis: verum

end;

end;