let G be Go-board; :: thesis: for i, j, m, n being 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 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 and
A2: i <= len G and
A3: 1 <= j and
A4: j < width G and
A5: 1 <= m and
A6: m <= len G and
A7: 1 <= n and
A8: n <= width G and
A9: p in cell (G,i,j) and
A10: p `1 = (G * (m,n)) `1 ; :: thesis: ( i = m or i = m -' 1 )
A11: (G * (m,1)) `1 = (G * (m,n)) `1 by A5, A6, A7, A8, GOBOARD5:2;
A12: 1 <= width G by A3, A4, XXREAL_0:2;
per cases ( i = len G or i < len G ) by A2, XXREAL_0:1;
suppose i = len G ; :: thesis: ( i = m or i = m -' 1 )
hence ( i = m or i = m -' 1 ) by A3, A4, A5, A6, A7, A8, A9, A10, Th21; :: 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, A3, A4, GOBRD11:32;
then consider r, s being Real such that
A13: p = |[r,s]| and
A14: (G * (i,1)) `1 <= r and
A15: r <= (G * ((i + 1),1)) `1 and
(G * (1,j)) `2 <= s and
s <= (G * (1,(j + 1))) `2 by A9;
A16: p `1 = r by A13, EUCLID:52;
( i <= m & m <= i + 1 )
proof
assume A17: ( not i <= m or not m <= i + 1 ) ; :: thesis: contradiction
end;
hence ( i = m or i = m -' 1 ) by Lm2; :: thesis: verum
end;
end;