let G be Go-board; :: thesis: for i, j, k, j1, k1 being 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 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 and

A2: i <= width G and

A3: 1 <= j and

A4: j <= j1 and

A5: j1 <= k1 and

A6: k1 <= k and

A7: k <= len G ; :: thesis: LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i)))

A8: j1 <= k by A5, A6, XXREAL_0:2;

j <= k1 by A4, A5, XXREAL_0:2;

then A9: 1 <= k1 by A3, XXREAL_0:2;

then A10: (G * (k1,i)) `1 <= (G * (k,i)) `1 by A1, A2, A6, A7, SPRECT_3:13;

A11: 1 <= j1 by A3, A4, XXREAL_0:2;

1 <= j1 by A3, A4, XXREAL_0:2;

then A12: 1 <= k by A8, XXREAL_0:2;

A13: k1 <= len G by A6, A7, XXREAL_0:2;

j <= k1 by A4, A5, XXREAL_0:2;

then A14: j <= len G by A13, XXREAL_0:2;

then (G * (j,i)) `2 = (G * (1,i)) `2 by A1, A2, A3, GOBOARD5:1

.= (G * (k,i)) `2 by A1, A2, A7, A12, GOBOARD5:1 ;

then A15: LSeg ((G * (j,i)),(G * (k,i))) is horizontal by SPPOL_1:15;

j1 <= k by A5, A6, XXREAL_0:2;

then A16: j1 <= len G by A7, XXREAL_0:2;

then A17: (G * (j,i)) `1 <= (G * (j1,i)) `1 by A1, A2, A3, A4, SPRECT_3:13;

A18: k1 <= len G by A6, A7, XXREAL_0:2;

then A19: (G * (j1,i)) `1 <= (G * (k1,i)) `1 by A1, A2, A5, A11, SPRECT_3:13;

(G * (j1,i)) `2 = (G * (1,i)) `2 by A1, A2, A11, A16, GOBOARD5:1

.= (G * (k1,i)) `2 by A1, A2, A9, A18, GOBOARD5:1 ;

then A20: LSeg ((G * (j1,i)),(G * (k1,i))) is horizontal by SPPOL_1:15;

(G * (j,i)) `2 = (G * (1,i)) `2 by A1, A2, A3, A14, GOBOARD5:1

.= (G * (j1,i)) `2 by A1, A2, A11, A16, GOBOARD5:1 ;

hence LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i))) by A15, A20, A17, A19, A10, GOBOARD7:64; :: thesis: verum

LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i)))

let i, j, k, j1, k1 be 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 and

A2: i <= width G and

A3: 1 <= j and

A4: j <= j1 and

A5: j1 <= k1 and

A6: k1 <= k and

A7: k <= len G ; :: thesis: LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i)))

A8: j1 <= k by A5, A6, XXREAL_0:2;

j <= k1 by A4, A5, XXREAL_0:2;

then A9: 1 <= k1 by A3, XXREAL_0:2;

then A10: (G * (k1,i)) `1 <= (G * (k,i)) `1 by A1, A2, A6, A7, SPRECT_3:13;

A11: 1 <= j1 by A3, A4, XXREAL_0:2;

1 <= j1 by A3, A4, XXREAL_0:2;

then A12: 1 <= k by A8, XXREAL_0:2;

A13: k1 <= len G by A6, A7, XXREAL_0:2;

j <= k1 by A4, A5, XXREAL_0:2;

then A14: j <= len G by A13, XXREAL_0:2;

then (G * (j,i)) `2 = (G * (1,i)) `2 by A1, A2, A3, GOBOARD5:1

.= (G * (k,i)) `2 by A1, A2, A7, A12, GOBOARD5:1 ;

then A15: LSeg ((G * (j,i)),(G * (k,i))) is horizontal by SPPOL_1:15;

j1 <= k by A5, A6, XXREAL_0:2;

then A16: j1 <= len G by A7, XXREAL_0:2;

then A17: (G * (j,i)) `1 <= (G * (j1,i)) `1 by A1, A2, A3, A4, SPRECT_3:13;

A18: k1 <= len G by A6, A7, XXREAL_0:2;

then A19: (G * (j1,i)) `1 <= (G * (k1,i)) `1 by A1, A2, A5, A11, SPRECT_3:13;

(G * (j1,i)) `2 = (G * (1,i)) `2 by A1, A2, A11, A16, GOBOARD5:1

.= (G * (k1,i)) `2 by A1, A2, A9, A18, GOBOARD5:1 ;

then A20: LSeg ((G * (j1,i)),(G * (k1,i))) is horizontal by SPPOL_1:15;

(G * (j,i)) `2 = (G * (1,i)) `2 by A1, A2, A3, A14, GOBOARD5:1

.= (G * (j1,i)) `2 by A1, A2, A11, A16, GOBOARD5:1 ;

hence LSeg ((G * (j1,i)),(G * (k1,i))) c= LSeg ((G * (j,i)),(G * (k,i))) by A15, A20, A17, A19, A10, GOBOARD7:64; :: thesis: verum