let i, j be Nat; :: thesis: for G being Go-board st 1 <= i & i <= len G & 1 <= j & j + 2 <= width G holds

(LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))}

let G be Go-board; :: thesis: ( 1 <= i & i <= len G & 1 <= j & j + 2 <= width G implies (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))} )

assume that

A1: ( 1 <= i & i <= len G ) and

A2: 1 <= j and

A3: j + 2 <= width G ; :: thesis: (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))}

(LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))}

let G be Go-board; :: thesis: ( 1 <= i & i <= len G & 1 <= j & j + 2 <= width G implies (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))} )

assume that

A1: ( 1 <= i & i <= len G ) and

A2: 1 <= j and

A3: j + 2 <= width G ; :: thesis: (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) = {(G * (i,(j + 1)))}

now :: thesis: for x being object holds

( ( x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2))))) 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,(j + 2))))) ) )

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

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

then ( x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) & x in LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))) ) by RLTOPSP1:68;

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

end;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,(j + 2))))) )

assume
x = G * (i,(j + 1))
; :: thesis: x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))))assume A4:
x in (LSeg ((G * (i,j)),(G * (i,(j + 1))))) /\ (LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))))
; :: thesis: x = G * (i,(j + 1))

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

A5: x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) by A4, XBOOLE_0:def 4;

A6: p in LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))) by A4, XBOOLE_0:def 4;

j <= j + 2 by NAT_1:11;

then A7: j <= width G by A3, XXREAL_0:2;

A8: j + 1 < j + 2 by XREAL_1:6;

then A9: j + 1 <= width G by A3, XXREAL_0:2;

A10: 1 <= j + 1 by NAT_1:11;

then (G * (i,(j + 1))) `1 = (G * (i,1)) `1 by A1, A9, GOBOARD5:2

.= (G * (i,j)) `1 by A1, A2, A7, GOBOARD5:2 ;

then A11: p `1 = (G * (i,(j + 1))) `1 by A5, Th5;

j < j + 1 by XREAL_1:29;

then (G * (i,j)) `2 < (G * (i,(j + 1))) `2 by A1, A2, A9, GOBOARD5:4;

then A12: p `2 <= (G * (i,(j + 1))) `2 by A5, TOPREAL1:4;

(G * (i,(j + 1))) `2 < (G * (i,(j + 2))) `2 by A1, A3, A8, A10, GOBOARD5:4;

then p `2 >= (G * (i,(j + 1))) `2 by A6, TOPREAL1:4;

then p `2 = (G * (i,(j + 1))) `2 by A12, XXREAL_0:1;

hence x = G * (i,(j + 1)) by A11, TOPREAL3:6; :: thesis: verum

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

A5: x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) by A4, XBOOLE_0:def 4;

A6: p in LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))) by A4, XBOOLE_0:def 4;

j <= j + 2 by NAT_1:11;

then A7: j <= width G by A3, XXREAL_0:2;

A8: j + 1 < j + 2 by XREAL_1:6;

then A9: j + 1 <= width G by A3, XXREAL_0:2;

A10: 1 <= j + 1 by NAT_1:11;

then (G * (i,(j + 1))) `1 = (G * (i,1)) `1 by A1, A9, GOBOARD5:2

.= (G * (i,j)) `1 by A1, A2, A7, GOBOARD5:2 ;

then A11: p `1 = (G * (i,(j + 1))) `1 by A5, Th5;

j < j + 1 by XREAL_1:29;

then (G * (i,j)) `2 < (G * (i,(j + 1))) `2 by A1, A2, A9, GOBOARD5:4;

then A12: p `2 <= (G * (i,(j + 1))) `2 by A5, TOPREAL1:4;

(G * (i,(j + 1))) `2 < (G * (i,(j + 2))) `2 by A1, A3, A8, A10, GOBOARD5:4;

then p `2 >= (G * (i,(j + 1))) `2 by A6, TOPREAL1:4;

then p `2 = (G * (i,(j + 1))) `2 by A12, XXREAL_0:1;

hence x = G * (i,(j + 1)) by A11, TOPREAL3:6; :: thesis: verum

then ( x in LSeg ((G * (i,j)),(G * (i,(j + 1)))) & x in LSeg ((G * (i,(j + 1))),(G * (i,(j + 2)))) ) by RLTOPSP1:68;

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