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

per cases ( i = len G or i < len G ) by A1, XXREAL_0:1;

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

then Int (v_strip G,i) = { |[r,s]| where r, s is Real : (G * i,1) `1 < r } by Th16;
then consider r, s being Real such that
A3: p = |[r,s]| and
A4: (G * i,1) `1 < r by A2;
thus p `1 > (G * i,1) `1 by A3, A4, EUCLID:56; :: thesis: verum

end;

suppose i < len G ; :: thesis: p `1 > (G * i,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, Th17;
then consider r, s being Real such that
A5: p = |[r,s]| and
A6: (G * i,1) `1 < r and
r < (G * (i + 1),1) `1 by A2;
thus p `1 > (G * i,1) `1 by A5, A6, EUCLID:56; :: thesis: verum

end;

end;