let i, j be 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 + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))}

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 + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))} )

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 + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))}

(LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))}

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 + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))} )

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 + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) = {(G * (i,j))}

now :: thesis: for x being object holds

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

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

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

then ( x in LSeg ((G * (i,j)),(G * ((i + 1),j))) & x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) ) by RLTOPSP1:68;

hence x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) by XBOOLE_0:def 4; :: thesis: verum

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

assume
x = G * (i,j)
; :: thesis: x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1)))))assume A5:
x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1)))))
; :: thesis: x = G * (i,j)

then reconsider p = x as Point of (TOP-REAL 2) ;

A6: x in LSeg ((G * (i,j)),(G * ((i + 1),j))) by A5, XBOOLE_0:def 4;

A7: p in LSeg ((G * (i,j)),(G * (i,(j + 1)))) by A5, XBOOLE_0:def 4;

A8: 1 <= i + 1 by NAT_1:11;

A9: 1 <= j + 1 by NAT_1:11;

j < j + 1 by XREAL_1:29;

then A10: j <= width G by A4, XXREAL_0:2;

i <= i + 1 by NAT_1:11;

then A11: i <= len G by A2, XXREAL_0:2;

then (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A3, A10, GOBOARD5:2

.= (G * (i,(j + 1))) `1 by A1, A4, A11, A9, GOBOARD5:2 ;

then A12: p `1 = (G * (i,j)) `1 by A7, Th5;

(G * (i,j)) `2 = (G * (1,j)) `2 by A1, A3, A11, A10, GOBOARD5:1

.= (G * ((i + 1),j)) `2 by A2, A3, A8, A10, GOBOARD5:1 ;

then p `2 = (G * (i,j)) `2 by A6, Th6;

hence x = G * (i,j) by A12, TOPREAL3:6; :: thesis: verum

end;then reconsider p = x as Point of (TOP-REAL 2) ;

A6: x in LSeg ((G * (i,j)),(G * ((i + 1),j))) by A5, XBOOLE_0:def 4;

A7: p in LSeg ((G * (i,j)),(G * (i,(j + 1)))) by A5, XBOOLE_0:def 4;

A8: 1 <= i + 1 by NAT_1:11;

A9: 1 <= j + 1 by NAT_1:11;

j < j + 1 by XREAL_1:29;

then A10: j <= width G by A4, XXREAL_0:2;

i <= i + 1 by NAT_1:11;

then A11: i <= len G by A2, XXREAL_0:2;

then (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A3, A10, GOBOARD5:2

.= (G * (i,(j + 1))) `1 by A1, A4, A11, A9, GOBOARD5:2 ;

then A12: p `1 = (G * (i,j)) `1 by A7, Th5;

(G * (i,j)) `2 = (G * (1,j)) `2 by A1, A3, A11, A10, GOBOARD5:1

.= (G * ((i + 1),j)) `2 by A2, A3, A8, A10, GOBOARD5:1 ;

then p `2 = (G * (i,j)) `2 by A6, Th6;

hence x = G * (i,j) by A12, TOPREAL3:6; :: thesis: verum

then ( x in LSeg ((G * (i,j)),(G * ((i + 1),j))) & x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) ) by RLTOPSP1:68;

hence x in (LSeg ((G * (i,j)),(G * ((i + 1),j)))) /\ (LSeg ((G * (i,j)),(G * (i,(j + 1))))) by XBOOLE_0:def 4; :: thesis: verum