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

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

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

then Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : r < (G * ((i + 1),1)) `1 } by Th12;
then ex r, s being Real st
( p = |[r,s]| & (G * ((i + 1),1)) `1 > r ) by A2;
hence p `1 < (G * ((i + 1),1)) `1 by EUCLID:52; :: thesis: verum

end;

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

then Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } by A1, Th14;
then ex r, s being Real st
( p = |[r,s]| & (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) by A2;
hence p `1 < (G * ((i + 1),1)) `1 by EUCLID:52; :: thesis: verum

end;

end;