let i, j be Element of NAT ; :: thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
(1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1))) = (1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),j))
let G be Go-board; :: thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies (1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1))) = (1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),j)) )
assume that
A1: 1 <= i and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: j + 1 <= width G ; :: thesis: (1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1))) = (1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),j))
A5: j < width G by A4, NAT_1:13;
A6: 1 <= j + 1 by NAT_1:11;
A7: 1 <= i + 1 by NAT_1:11;
then A8: (G * (i + 1),j) `1 = (G * (i + 1),1) `1 by A2, A3, A5, GOBOARD5:3
.= (G * (i + 1),(j + 1)) `1 by A2, A4, A7, A6, GOBOARD5:3 ;
A9: i < len G by A2, NAT_1:13;
then A10: (G * i,j) `1 = (G * i,1) `1 by A1, A3, A5, GOBOARD5:3
.= (G * i,(j + 1)) `1 by A1, A4, A9, A6, GOBOARD5:3 ;
A11: (G * (i + 1),(j + 1)) `2 = (G * 1,(j + 1)) `2 by A2, A4, A7, A6, GOBOARD5:2
.= (G * i,(j + 1)) `2 by A1, A4, A9, A6, GOBOARD5:2 ;
A12: (G * i,j) `2 = (G * 1,j) `2 by A1, A3, A9, A5, GOBOARD5:2
.= (G * (i + 1),j) `2 by A2, A3, A7, A5, GOBOARD5:2 ;
A13: ((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))) `2 = (1 / 2) * (((G * i,j) + (G * (i + 1),(j + 1))) `2 ) by TOPREAL3:9
.= (1 / 2) * (((G * i,j) `2 ) + ((G * (i + 1),(j + 1)) `2 )) by TOPREAL3:7
.= (1 / 2) * (((G * i,(j + 1)) + (G * (i + 1),j)) `2 ) by A12, A11, TOPREAL3:7
.= ((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),j))) `2 by TOPREAL3:9 ;
((1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1)))) `1 = (1 / 2) * (((G * i,j) + (G * (i + 1),(j + 1))) `1 ) by TOPREAL3:9
.= (1 / 2) * (((G * i,j) `1 ) + ((G * (i + 1),(j + 1)) `1 )) by TOPREAL3:7
.= (1 / 2) * (((G * i,(j + 1)) + (G * (i + 1),j)) `1 ) by A10, A8, TOPREAL3:7
.= ((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),j))) `1 by TOPREAL3:9 ;
hence (1 / 2) * ((G * i,j) + (G * (i + 1),(j + 1))) = |[(((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),j))) `1 ),(((1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),j))) `2 )]| by A13, EUCLID:57
.= (1 / 2) * ((G * i,(j + 1)) + (G * (i + 1),j)) by EUCLID:57 ;
:: thesis: verum