let i, j be Element of NAT ; :: thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
(LSeg (G * i,j),(G * i,(j + 1))) /\ (LSeg (G * i,(j + 1)),(G * (i + 1),(j + 1))) = {(G * i,(j + 1))}
let G be Go-board; :: thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies (LSeg (G * i,j),(G * i,(j + 1))) /\ (LSeg (G * i,(j + 1)),(G * (i + 1),(j + 1))) = {(G * i,(j + 1))} )
assume that
A1: 1 <= i and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: j + 1 <= width G ; :: thesis: (LSeg (G * i,j),(G * i,(j + 1))) /\ (LSeg (G * i,(j + 1)),(G * (i + 1),(j + 1))) = {(G * i,(j + 1))}

now

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

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

assume A5: x in (LSeg (G * i,j),(G * i,(j + 1))) /\ (LSeg (G * i,(j + 1)),(G * (i + 1),(j + 1))) ; :: thesis: x = G * i,(j + 1)
then reconsider p = x as Point of (TOP-REAL 2) ;
A6: x in LSeg (G * i,j),(G * i,(j + 1)) by A5, XBOOLE_0:def 4;
A7: p in LSeg (G * i,(j + 1)),(G * (i + 1),(j + 1)) by A5, XBOOLE_0:def 4;
A8: 1 <= i + 1 by NAT_1:11;
i <= i + 1 by NAT_1:11;
then A9: i <= len G by A2, XXREAL_0:2;
A10: 1 <= j + 1 by NAT_1:11;
then (G * i,(j + 1)) `2 = (G * 1,(j + 1)) `2 by A1, A4, A9, GOBOARD5:2
.= (G * (i + 1),(j + 1)) `2 by A2, A4, A10, A8, GOBOARD5:2 ;
then A11: p `2 = (G * i,(j + 1)) `2 by A7, Th6;
j < j + 1 by XREAL_1:31;
then j <= width G by A4, XXREAL_0:2;
then (G * i,j) `1 = (G * i,1) `1 by A1, A3, A9, GOBOARD5:3
.= (G * i,(j + 1)) `1 by A1, A4, A9, A10, GOBOARD5:3 ;
then p `1 = (G * i,(j + 1)) `1 by A6, Th5;
hence x = G * i,(j + 1) by A11, TOPREAL3:11; :: thesis: verum

end;

assume x = G * i,(j + 1) ; :: thesis: x in (LSeg (G * i,j),(G * i,(j + 1))) /\ (LSeg (G * i,(j + 1)),(G * (i + 1),(j + 1)))
then ( x in LSeg (G * i,j),(G * i,(j + 1)) & x in LSeg (G * i,(j + 1)),(G * (i + 1),(j + 1)) ) by RLTOPSP1:69;
hence x in (LSeg (G * i,j),(G * i,(j + 1))) /\ (LSeg (G * i,(j + 1)),(G * (i + 1),(j + 1))) by XBOOLE_0:def 4; :: thesis: verum

end;

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