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 `2 = (G * (m,n)) `2 & not j = n holds

j = n -' 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 `2 = (G * (m,n)) `2 & not j = n holds

j = n -' 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 `2 = (G * (m,n)) `2 & not j = n implies j = n -' 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 `2 = (G * (m,n)) `2 ; :: thesis: ( j = n or j = n -' 1 )

A11: (G * (1,n)) `2 = (G * (m,n)) `2 by A5, A6, A7, A8, GOBOARD5:1;

A12: 1 <= len G by A1, A2, XXREAL_0:2;

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 `2 = (G * (m,n)) `2 & not j = n holds

j = n -' 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 `2 = (G * (m,n)) `2 & not j = n holds

j = n -' 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 `2 = (G * (m,n)) `2 & not j = n implies j = n -' 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 `2 = (G * (m,n)) `2 ; :: thesis: ( j = n or j = n -' 1 )

A11: (G * (1,n)) `2 = (G * (m,n)) `2 by A5, A6, A7, A8, GOBOARD5:1;

A12: 1 <= len G by A1, A2, XXREAL_0:2;

per cases
( j = width G or j < width G )
by A4, XXREAL_0:1;

end;

suppose
j < width G
; :: thesis: ( j = n or j = n -' 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, A3, GOBRD11:32;

then consider r, s being Real such that

A13: p = |[r,s]| and

(G * (i,1)) `1 <= r and

r <= (G * ((i + 1),1)) `1 and

A14: (G * (1,j)) `2 <= s and

A15: s <= (G * (1,(j + 1))) `2 by A9;

A16: p `2 = s by A13, EUCLID:52;

( j <= n & n <= j + 1 )

end;then consider r, s being Real such that

A13: p = |[r,s]| and

(G * (i,1)) `1 <= r and

r <= (G * ((i + 1),1)) `1 and

A14: (G * (1,j)) `2 <= s and

A15: s <= (G * (1,(j + 1))) `2 by A9;

A16: p `2 = s by A13, EUCLID:52;

( j <= n & n <= j + 1 )

proof

hence
( j = n or j = n -' 1 )
by Lm2; :: thesis: verum
assume A17:
( not j <= n or not n <= j + 1 )
; :: thesis: contradiction

end;