let G be Go-board; :: thesis: for i, j, k, j1, k1 being Element of NAT st 1 <= i & i <= len G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G holds
LSeg (G * i,j1),(G * i,k1) c= LSeg (G * i,j),(G * i,k)
let i, j, k, j1, k1 be Element of NAT ; :: thesis: ( 1 <= i & i <= len G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G implies LSeg (G * i,j1),(G * i,k1) c= LSeg (G * i,j),(G * i,k) )
assume that
A1: ( 1 <= i & i <= len G ) and
A2: ( 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G ) ; :: thesis: LSeg (G * i,j1),(G * i,k1) c= LSeg (G * i,j),(G * i,k)
( j <= k1 & k1 <= width G & 1 <= j1 & j1 <= k ) by A2, XXREAL_0:2;
then A3: ( j <= width G & 1 <= k ) by XXREAL_0:2;
then (G * i,j) `1 = (G * i,1) `1 by A1, A2, GOBOARD5:3
.= (G * i,k) `1 by A1, A2, A3, GOBOARD5:3 ;
then A4: LSeg (G * i,j),(G * i,k) is vertical by SPPOL_1:37;
( j1 <= k & j <= k1 ) by A2, XXREAL_0:2;
then A5: ( 1 <= j1 & j1 <= width G & 1 <= k1 & k1 <= width G ) by A2, XXREAL_0:2;
then (G * i,j1) `1 = (G * i,1) `1 by A1, GOBOARD5:3
.= (G * i,k1) `1 by A1, A5, GOBOARD5:3 ;
then A6: LSeg (G * i,j1),(G * i,k1) is vertical by SPPOL_1:37;
A7: (G * i,j) `1 = (G * i,1) `1 by A1, A2, A3, GOBOARD5:3
.= (G * i,j1) `1 by A1, A5, GOBOARD5:3 ;
A8: (G * i,j) `2 <= (G * i,j1) `2 by A1, A2, A5, SPRECT_3:24;
A9: (G * i,j1) `2 <= (G * i,k1) `2 by A1, A2, A5, SPRECT_3:24;
(G * i,k1) `2 <= (G * i,k) `2 by A1, A2, A5, SPRECT_3:24;
hence LSeg (G * i,j1),(G * i,k1) c= LSeg (G * i,j),(G * i,k) by A4, A6, A7, A8, A9, GOBOARD7:65; :: thesis: verum