let i1, j1, i2, j2 be Element of NAT ; :: thesis: for G being Go-board st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & (1 / 2) * ((G * i1,j1) + (G * i1,(j1 + 1))) in LSeg (G * i2,j2),(G * i2,(j2 + 1)) holds
( i1 = i2 & j1 = j2 )
let G be Go-board; :: thesis: ( 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2 + 1 <= width G & (1 / 2) * ((G * i1,j1) + (G * i1,(j1 + 1))) in LSeg (G * i2,j2),(G * i2,(j2 + 1)) implies ( i1 = i2 & j1 = j2 ) )
assume that
A1: ( 1 <= i1 & i1 <= len G ) and
A2: 1 <= j1 and
A3: j1 + 1 <= width G and
A4: ( 1 <= i2 & i2 <= len G ) and
A5: 1 <= j2 and
A6: j2 + 1 <= width G ; :: thesis: ( not (1 / 2) * ((G * i1,j1) + (G * i1,(j1 + 1))) in LSeg (G * i2,j2),(G * i2,(j2 + 1)) or ( i1 = i2 & j1 = j2 ) )
set mi = (1 / 2) * ((G * i1,j1) + (G * i1,(j1 + 1)));
A7: ((1 / 2) * (G * i1,j1)) + ((1 / 2) * (G * i1,(j1 + 1))) = (1 / 2) * ((G * i1,j1) + (G * i1,(j1 + 1))) by EUCLID:36;
then A8: (1 / 2) * ((G * i1,j1) + (G * i1,(j1 + 1))) in LSeg (G * i1,j1),(G * i1,(j1 + 1)) by Lm1;
assume A9: (1 / 2) * ((G * i1,j1) + (G * i1,(j1 + 1))) in LSeg (G * i2,j2),(G * i2,(j2 + 1)) ; :: thesis: ( i1 = i2 & j1 = j2 )
then A10: LSeg (G * i1,j1),(G * i1,(j1 + 1)) meets LSeg (G * i2,j2),(G * i2,(j2 + 1)) by A8, XBOOLE_0:3;
hence A11: i1 = i2 by A1, A2, A3, A4, A5, A6, Th21; :: thesis: j1 = j2

now

j1 < j1 + 1 by XREAL_1:31;
then A12: (G * i1,(j1 + 1)) `2 > (G * i1,j1) `2 by A1, A2, A3, GOBOARD5:5;
assume A13: abs (j1 - j2) = 1 ; :: thesis: contradiction

per cases ( j1 = j2 + 1 or j1 + 1 = j2 ) by A13, SEQM_3:81;

suppose A14: j1 = j2 + 1 ; :: thesis: contradiction

then (LSeg (G * i2,j2),(G * i2,(j2 + 1))) /\ (LSeg (G * i2,(j2 + 1)),(G * i2,(j2 + 2))) = {(G * i2,(j2 + 1))} by A3, A4, A5, Th15;
then (1 / 2) * ((G * i1,j1) + (G * i1,(j1 + 1))) in {(G * i1,j1)} by A9, A8, A11, A14, XBOOLE_0:def 4;
then ((1 / 2) * (G * i1,j1)) + ((1 / 2) * (G * i1,(j1 + 1))) = G * i1,j1 by A7, TARSKI:def 1
.= ((1 / 2) + (1 / 2)) * (G * i1,j1) by EUCLID:33
.= ((1 / 2) * (G * i1,j1)) + ((1 / 2) * (G * i1,j1)) by EUCLID:37 ;
then (1 / 2) * (G * i1,j1) = (1 / 2) * (G * i1,(j1 + 1)) by Th3;
hence contradiction by A12, EUCLID:38; :: thesis: verum

end;

suppose A15: j1 + 1 = j2 ; :: thesis: contradiction

then (LSeg (G * i2,j1),(G * i2,(j1 + 1))) /\ (LSeg (G * i2,(j1 + 1)),(G * i2,(j1 + 2))) = {(G * i2,(j1 + 1))} by A2, A4, A6, Th15;
then (1 / 2) * ((G * i1,j1) + (G * i1,(j1 + 1))) in {(G * i1,j2)} by A9, A8, A11, A15, XBOOLE_0:def 4;
then ((1 / 2) * (G * i1,j1)) + ((1 / 2) * (G * i1,(j1 + 1))) = G * i1,j2 by A7, TARSKI:def 1
.= ((1 / 2) + (1 / 2)) * (G * i1,j2) by EUCLID:33
.= ((1 / 2) * (G * i1,j2)) + ((1 / 2) * (G * i1,j2)) by EUCLID:37 ;
then (1 / 2) * (G * i1,j1) = (1 / 2) * (G * i1,(j1 + 1)) by A15, Th3;
hence contradiction by A12, EUCLID:38; :: thesis: verum

end;

end;

end;

then abs (j1 - j2) = 0 by A1, A2, A3, A4, A5, A6, A10, Th21, NAT_1:26;
hence j1 = j2 by Th2; :: thesis: verum