let i, j be Element of NAT ; :: thesis: for G being Go-board
for p being Point of (TOP-REAL 2) st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell G,i,j holds
p is_extremal_in cell G,i,j
let G be Go-board; :: thesis: for p being Point of (TOP-REAL 2) st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell G,i,j holds
p is_extremal_in cell G,i,j
let p be Point of (TOP-REAL 2); :: thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell G,i,j implies p is_extremal_in cell G,i,j )
assume that
A1: 1 <= i and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: j + 1 <= width G and
A5: p in Values G and
A6: p in cell G,i,j ; :: thesis: p is_extremal_in cell G,i,j
for a, b being Point of (TOP-REAL 2) st p in LSeg a,b & LSeg a,b c= cell G,i,j & not p = a holds
p = b

proof

let a, b be Point of (TOP-REAL 2); :: thesis: ( p in LSeg a,b & LSeg a,b c= cell G,i,j & not p = a implies p = b )
assume that
A7: p in LSeg a,b and
A8: LSeg a,b c= cell G,i,j ; :: thesis: ( p = a or p = b )
assume A9: ( a <> p & b <> p ) ; :: thesis: contradiction
A10: ( a in LSeg a,b & b in LSeg a,b ) by RLTOPSP1:69;

per cases ( p = G * i,j or p = G * i,(j + 1) or p = G * (i + 1),(j + 1) or p = G * (i + 1),j ) by A1, A2, A3, A4, A5, A6, Th21;

suppose A11: p = G * i,j ; :: thesis: contradiction

then A12: ( p `1 <= a `1 & p `2 <= a `2 ) by A1, A2, A3, A4, A8, A10, Th19;
A13: ( p `1 <= b `1 & p `2 <= b `2 ) by A1, A2, A3, A4, A8, A10, A11, Th19;

now

per cases ( ( a `1 <= b `1 & a `2 <= b `2 ) or ( a `1 <= b `1 & b `2 < a `2 ) or ( b `1 < a `1 & a `2 <= b `2 ) or ( b `1 < a `1 & b `2 < a `2 ) ) ;

suppose ( a `1 <= b `1 & a `2 <= b `2 ) ; :: thesis: contradiction

then ( a `1 <= p `1 & a `2 <= p `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then ( a `1 = p `1 & a `2 = p `2 ) by A12, XXREAL_0:1;
hence contradiction by A9, TOPREAL3:11; :: thesis: verum

end;

suppose ( a `1 <= b `1 & b `2 < a `2 ) ; :: thesis: contradiction

then ( a `1 <= p `1 & b `2 <= p `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then A14: ( a `1 = p `1 & b `2 = p `2 ) by A12, A13, XXREAL_0:1;
then a `2 <> p `2 by A9, TOPREAL3:11;
then LSeg a,b is vertical by A7, A10, A14, SPPOL_1:39;
then a `1 = b `1 by SPPOL_1:37;
hence contradiction by A9, A14, TOPREAL3:11; :: thesis: verum

end;

suppose ( b `1 < a `1 & a `2 <= b `2 ) ; :: thesis: contradiction

then ( b `1 <= p `1 & a `2 <= p `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then A15: ( b `1 = p `1 & a `2 = p `2 ) by A12, A13, XXREAL_0:1;
then b `2 <> p `2 by A9, TOPREAL3:11;
then LSeg a,b is vertical by A7, A10, A15, SPPOL_1:39;
then a `1 = b `1 by SPPOL_1:37;
hence contradiction by A9, A15, TOPREAL3:11; :: thesis: verum

end;

suppose ( b `1 < a `1 & b `2 < a `2 ) ; :: thesis: contradiction

then ( b `1 <= p `1 & b `2 <= p `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then ( b `1 = p `1 & b `2 = p `2 ) by A13, XXREAL_0:1;
hence contradiction by A9, TOPREAL3:11; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum

end;

suppose A16: p = G * i,(j + 1) ; :: thesis: contradiction

then A17: p `1 = (G * i,j) `1 by A1, A2, A3, A4, Th18;
then A18: ( p `1 <= a `1 & a `2 <= p `2 ) by A1, A2, A3, A4, A8, A10, A16, Th19;
A19: ( p `1 <= b `1 & b `2 <= p `2 ) by A1, A2, A3, A4, A8, A10, A16, A17, Th19;

now

per cases ( ( a `1 <= b `1 & a `2 <= b `2 ) or ( a `1 <= b `1 & b `2 < a `2 ) or ( b `1 < a `1 & a `2 <= b `2 ) or ( b `1 < a `1 & b `2 < a `2 ) ) ;

suppose ( a `1 <= b `1 & a `2 <= b `2 ) ; :: thesis: contradiction

then ( a `1 <= p `1 & p `2 <= b `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then A20: ( a `1 = p `1 & b `2 = p `2 ) by A18, A19, XXREAL_0:1;
then a `2 <> p `2 by A9, TOPREAL3:11;
then LSeg a,b is vertical by A7, A10, A20, SPPOL_1:39;
then a `1 = b `1 by SPPOL_1:37;
hence contradiction by A9, A20, TOPREAL3:11; :: thesis: verum

end;

suppose ( a `1 <= b `1 & b `2 < a `2 ) ; :: thesis: contradiction

then ( a `1 <= p `1 & p `2 <= a `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then ( a `1 = p `1 & a `2 = p `2 ) by A18, XXREAL_0:1;
hence contradiction by A9, TOPREAL3:11; :: thesis: verum

end;

suppose ( b `1 < a `1 & a `2 <= b `2 ) ; :: thesis: contradiction

then ( b `1 <= p `1 & p `2 <= b `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then ( b `1 = p `1 & b `2 = p `2 ) by A19, XXREAL_0:1;
hence contradiction by A9, TOPREAL3:11; :: thesis: verum

end;

suppose ( b `1 < a `1 & b `2 < a `2 ) ; :: thesis: contradiction

then ( b `1 <= p `1 & p `2 <= a `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then A21: ( b `1 = p `1 & a `2 = p `2 ) by A18, A19, XXREAL_0:1;
then b `2 <> p `2 by A9, TOPREAL3:11;
then LSeg a,b is vertical by A7, A10, A21, SPPOL_1:39;
then a `1 = b `1 by SPPOL_1:37;
hence contradiction by A9, A21, TOPREAL3:11; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum

end;

suppose p = G * (i + 1),(j + 1) ; :: thesis: contradiction

then A22: ( p `1 = (G * (i + 1),j) `1 & p `2 = (G * i,(j + 1)) `2 ) by A1, A2, A3, A4, Th18;
then A23: ( a `1 <= p `1 & a `2 <= p `2 ) by A1, A2, A3, A4, A8, A10, Th19;
A24: ( b `1 <= p `1 & b `2 <= p `2 ) by A1, A2, A3, A4, A8, A10, A22, Th19;

now

per cases ( ( a `1 <= b `1 & a `2 <= b `2 ) or ( a `1 <= b `1 & b `2 < a `2 ) or ( b `1 < a `1 & a `2 <= b `2 ) or ( b `1 < a `1 & b `2 < a `2 ) ) ;

suppose ( a `1 <= b `1 & a `2 <= b `2 ) ; :: thesis: contradiction

then ( p `1 <= b `1 & p `2 <= b `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then ( b `1 = p `1 & b `2 = p `2 ) by A24, XXREAL_0:1;
hence contradiction by A9, TOPREAL3:11; :: thesis: verum

end;

suppose ( a `1 <= b `1 & b `2 < a `2 ) ; :: thesis: contradiction

then ( p `1 <= b `1 & p `2 <= a `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then A25: ( b `1 = p `1 & a `2 = p `2 ) by A23, A24, XXREAL_0:1;
then b `2 <> p `2 by A9, TOPREAL3:11;
then LSeg a,b is vertical by A7, A10, A25, SPPOL_1:39;
then a `1 = b `1 by SPPOL_1:37;
hence contradiction by A9, A25, TOPREAL3:11; :: thesis: verum

end;

suppose ( b `1 < a `1 & a `2 <= b `2 ) ; :: thesis: contradiction

then ( p `1 <= a `1 & p `2 <= b `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then A26: ( a `1 = p `1 & b `2 = p `2 ) by A23, A24, XXREAL_0:1;
then a `2 <> p `2 by A9, TOPREAL3:11;
then LSeg a,b is vertical by A7, A10, A26, SPPOL_1:39;
then a `1 = b `1 by SPPOL_1:37;
hence contradiction by A9, A26, TOPREAL3:11; :: thesis: verum

end;

suppose ( b `1 < a `1 & b `2 < a `2 ) ; :: thesis: contradiction

then ( p `1 <= a `1 & p `2 <= a `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then ( a `1 = p `1 & a `2 = p `2 ) by A23, XXREAL_0:1;
hence contradiction by A9, TOPREAL3:11; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum

end;

suppose A27: p = G * (i + 1),j ; :: thesis: contradiction

then A28: p `2 = (G * i,j) `2 by A1, A2, A3, A4, Th18;
then A29: ( a `1 <= p `1 & p `2 <= a `2 ) by A1, A2, A3, A4, A8, A10, A27, Th19;
A30: ( b `1 <= p `1 & p `2 <= b `2 ) by A1, A2, A3, A4, A8, A10, A27, A28, Th19;

now

per cases ( ( a `1 <= b `1 & a `2 <= b `2 ) or ( a `1 <= b `1 & b `2 < a `2 ) or ( b `1 < a `1 & a `2 <= b `2 ) or ( b `1 < a `1 & b `2 < a `2 ) ) ;

suppose ( a `1 <= b `1 & a `2 <= b `2 ) ; :: thesis: contradiction

then ( p `1 <= b `1 & a `2 <= p `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then A31: ( b `1 = p `1 & a `2 = p `2 ) by A29, A30, XXREAL_0:1;
then b `2 <> p `2 by A9, TOPREAL3:11;
then LSeg a,b is vertical by A7, A10, A31, SPPOL_1:39;
then a `1 = b `1 by SPPOL_1:37;
hence contradiction by A9, A31, TOPREAL3:11; :: thesis: verum

end;

suppose ( a `1 <= b `1 & b `2 < a `2 ) ; :: thesis: contradiction

then ( p `1 <= b `1 & b `2 <= p `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then ( b `1 = p `1 & b `2 = p `2 ) by A30, XXREAL_0:1;
hence contradiction by A9, TOPREAL3:11; :: thesis: verum

end;

suppose ( b `1 < a `1 & a `2 <= b `2 ) ; :: thesis: contradiction

then ( p `1 <= a `1 & a `2 <= p `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then ( a `1 = p `1 & a `2 = p `2 ) by A29, XXREAL_0:1;
hence contradiction by A9, TOPREAL3:11; :: thesis: verum

end;

suppose ( b `1 < a `1 & b `2 < a `2 ) ; :: thesis: contradiction

then ( p `1 <= a `1 & b `2 <= p `2 ) by A7, TOPREAL1:9, TOPREAL1:10;
then A32: ( a `1 = p `1 & b `2 = p `2 ) by A29, A30, XXREAL_0:1;
then a `2 <> p `2 by A9, TOPREAL3:11;
then LSeg a,b is vertical by A7, A10, A32, SPPOL_1:39;
then a `1 = b `1 by SPPOL_1:37;
hence contradiction by A9, A32, TOPREAL3:11; :: thesis: verum

end;

end;

end;

hence contradiction ; :: thesis: verum

end;

end;

end;

hence p is_extremal_in cell G,i,j by A6, SPPOL_1:def 1; :: thesis: verum