let i, j be Nat; :: thesis: for G being Go-board st 1 <= i & i + 2 <= len G & 1 <= j & j <= width G holds
(LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) = {(G * ((i + 1),j))}
let G be Go-board; :: thesis: ( 1 <= i & i + 2 <= len G & 1 <= j & j <= width G implies (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) = {(G * ((i + 1),j))} )
assume that
A1: 1 <= i and
A2: i + 2 <= len G and
A3: ( 1 <= j & j <= width G ) ; :: thesis: (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) = {(G * ((i + 1),j))}

now :: thesis: for x being object holds
( ( x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) implies x = G * ((i + 1),j) ) & ( x = G * ((i + 1),j) implies x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) ) )

let x be object ; :: thesis: ( ( x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) implies x = G * ((i + 1),j) ) & ( x = G * ((i + 1),j) implies x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) ) )

hereby :: thesis: ( x = G * ((i + 1),j) implies x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) )

assume A4: x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) ; :: thesis: x = G * ((i + 1),j)
then reconsider p = x as Point of (TOP-REAL 2) ;
A5: x in LSeg ((G * (i,j)),(G * ((i + 1),j))) by A4, XBOOLE_0:def 4;
A6: p in LSeg ((G * ((i + 1),j)),(G * ((i + 2),j))) by A4, XBOOLE_0:def 4;
i <= i + 2 by NAT_1:11;
then A7: i <= len G by A2, XXREAL_0:2;
A8: i + 1 < i + 2 by XREAL_1:6;
then A9: i + 1 <= len G by A2, XXREAL_0:2;
A10: 1 <= i + 1 by NAT_1:11;
then (G * ((i + 1),j)) `2 = (G * (1,j)) `2 by A3, A9, GOBOARD5:1
.= (G * (i,j)) `2 by A1, A3, A7, GOBOARD5:1 ;
then A11: p `2 = (G * ((i + 1),j)) `2 by A5, Th6;
i < i + 1 by XREAL_1:29;
then (G * (i,j)) `1 < (G * ((i + 1),j)) `1 by A1, A3, A9, GOBOARD5:3;
then A12: p `1 <= (G * ((i + 1),j)) `1 by A5, TOPREAL1:3;
(G * ((i + 1),j)) `1 < (G * ((i + 2),j)) `1 by A2, A3, A8, A10, GOBOARD5:3;
then p `1 >= (G * ((i + 1),j)) `1 by A6, TOPREAL1:3;
then p `1 = (G * ((i + 1),j)) `1 by A12, XXREAL_0:1;
hence x = G * ((i + 1),j) by A11, TOPREAL3:6; :: thesis: verum

end;

assume x = G * ((i + 1),j) ; :: thesis: x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j))))
then ( x in LSeg ((G * (i,j)),(G * ((i + 1),j))) & x in LSeg ((G * ((i + 1),j)),(G * ((i + 2),j))) ) by RLTOPSP1:68;
hence x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) by XBOOLE_0:def 4; :: thesis: verum

end;

hence (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * ((i + 1),j)),(G * ((i + 2),j)))) = {(G * ((i + 1),j))} by TARSKI:def 1; :: thesis: verum