let j be 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 Th15;
then ex r, s being Real st
( p = |[r,s]| & (G * (1,(j + 1))) `2 > s ) by A2;
hence p `2 < (G * (1,(j + 1))) `2 by EUCLID:52; :: 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, Th17;
then ex r, s being Real st
( p = |[r,s]| & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) by A2;
hence p `2 < (G * (1,(j + 1))) `2 by EUCLID:52; :: thesis: verum

end;

end;