let G be Go-board; :: thesis: for i, j, m, n being Element of NAT
for p being Point of (TOP-REAL 2) st 1 <= i & i <= len G & 1 <= j & j < width G & 1 <= m & m <= len G & 1 <= n & n <= width G & p in cell G,i,j & p `1 = (G * m,n) `1 & not i = m holds
i = m -' 1
let i, j, m, n be Element of NAT ; :: thesis: for p being Point of (TOP-REAL 2) st 1 <= i & i <= len G & 1 <= j & j < width G & 1 <= m & m <= len G & 1 <= n & n <= width G & p in cell G,i,j & p `1 = (G * m,n) `1 & not i = m holds
i = m -' 1
let p be Point of (TOP-REAL 2); :: thesis: ( 1 <= i & i <= len G & 1 <= j & j < width G & 1 <= m & m <= len G & 1 <= n & n <= width G & p in cell G,i,j & p `1 = (G * m,n) `1 & not i = m implies i = m -' 1 )
assume that
A1: ( 1 <= i & i <= len G ) and
A2: ( 1 <= j & j < width G ) and
A3: ( 1 <= m & m <= len G ) and
A4: ( 1 <= n & n <= width G ) and
A5: p in cell G,i,j and
A6: p `1 = (G * m,n) `1 ; :: thesis: ( i = m or i = m -' 1 )
A7: 1 <= width G by A2, XXREAL_0:2;
A8: (G * m,1) `1 = (G * m,n) `1 by A3, A4, GOBOARD5:3;
per cases ( i = len G or i < len G ) by A1, XXREAL_0:1;
suppose i = len G ; :: thesis: ( i = m or i = m -' 1 )
hence ( i = m or i = m -' 1 ) by A2, A3, A4, A5, A6, Th22; :: thesis: verum
end;
suppose i < len G ; :: thesis: ( i = m or i = m -' 1 )
then cell G,i,j = { |[r,s]| where r, s is Real : ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } by A1, A2, GOBRD11:32;
then consider r, s being Real such that
A9: p = |[r,s]| and
A10: ( (G * i,1) `1 <= r & r <= (G * (i + 1),1) `1 ) and
( (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) by A5;
A11: p `1 = r by A9, EUCLID:56;
( i <= m & m <= i + 1 )
proof
assume A12: ( not i <= m or not m <= i + 1 ) ; :: thesis: contradiction
end;
hence ( i = m or i = m -' 1 ) by Lm2; :: thesis: verum
end;
end;