let G be Go-board; :: thesis: for i, j, k, j1, k1 being Element of NAT st 1 <= i & i <= width G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G holds
LSeg (G * j1,i),(G * k1,i) c= LSeg (G * j,i),(G * k,i)
let i, j, k, j1, k1 be Element of NAT ; :: thesis: ( 1 <= i & i <= width G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G implies LSeg (G * j1,i),(G * k1,i) c= LSeg (G * j,i),(G * k,i) )
assume that
A1: ( 1 <= i & i <= width G ) and
A2: ( 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G ) ; :: thesis: LSeg (G * j1,i),(G * k1,i) c= LSeg (G * j,i),(G * k,i)
( j <= k1 & k1 <= len G & 1 <= j1 & j1 <= k ) by A2, XXREAL_0:2;
then A3: ( j <= len G & 1 <= k ) by XXREAL_0:2;
then (G * j,i) `2 = (G * 1,i) `2 by A1, A2, GOBOARD5:2
.= (G * k,i) `2 by A1, A2, A3, GOBOARD5:2 ;
then A4: LSeg (G * j,i),(G * k,i) is horizontal by SPPOL_1:36;
( j1 <= k & j <= k1 ) by A2, XXREAL_0:2;
then A5: ( 1 <= j1 & j1 <= len G & 1 <= k1 & k1 <= len G ) by A2, XXREAL_0:2;
then (G * j1,i) `2 = (G * 1,i) `2 by A1, GOBOARD5:2
.= (G * k1,i) `2 by A1, A5, GOBOARD5:2 ;
then A6: LSeg (G * j1,i),(G * k1,i) is horizontal by SPPOL_1:36;
A7: (G * j,i) `2 = (G * 1,i) `2 by A1, A2, A3, GOBOARD5:2
.= (G * j1,i) `2 by A1, A5, GOBOARD5:2 ;
A8: (G * j,i) `1 <= (G * j1,i) `1 by A1, A2, A5, SPRECT_3:25;
A9: (G * j1,i) `1 <= (G * k1,i) `1 by A1, A2, A5, SPRECT_3:25;
(G * k1,i) `1 <= (G * k,i) `1 by A1, A2, A5, SPRECT_3:25;
hence LSeg (G * j1,i),(G * k1,i) c= LSeg (G * j,i),(G * k,i) by A4, A6, A7, A8, A9, GOBOARD7:66; :: thesis: verum