let i, j be Element of 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

let x be set ; :: 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_1:8;
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, GOBOARD5:2
.= (G * i,j) `2 by A1, A3, A7, GOBOARD5:2 ;
then A11: p `2 = (G * (i + 1),j) `2 by A5, Th6;
i < i + 1 by XREAL_1:31;
then (G * i,j) `1 < (G * (i + 1),j) `1 by A1, A3, A9, GOBOARD5:4;
then A12: p `1 <= (G * (i + 1),j) `1 by A5, TOPREAL1:9;
(G * (i + 1),j) `1 < (G * (i + 2),j) `1 by A2, A3, A8, A10, GOBOARD5:4;
then p `1 >= (G * (i + 1),j) `1 by A6, TOPREAL1:9;
then p `1 = (G * (i + 1),j) `1 by A12, XXREAL_0:1;
hence x = G * (i + 1),j by A11, TOPREAL3:11; :: 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 RLTOPSP1:69;
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