let i1, i2, j1, j2 be 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 RLVECT_1:def 5;

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, Th19; :: thesis: j1 = j2

hence j1 = j2 by Th2; :: thesis: verum

( 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 RLVECT_1:def 5;

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, Th19; :: thesis: j1 = j2

now :: thesis: not |.(j1 - j2).| = 1

then
|.(j1 - j2).| = 0
by A1, A2, A3, A4, A5, A6, A10, Th19, NAT_1:25;
j1 < j1 + 1
by XREAL_1:29;

then A12: (G * (i1,(j1 + 1))) `2 > (G * (i1,j1)) `2 by A1, A2, A3, GOBOARD5:4;

assume A13: |.(j1 - j2).| = 1 ; :: thesis: contradiction

end;then A12: (G * (i1,(j1 + 1))) `2 > (G * (i1,j1)) `2 by A1, A2, A3, GOBOARD5:4;

assume A13: |.(j1 - j2).| = 1 ; :: thesis: contradiction

per cases
( j1 = j2 + 1 or j1 + 1 = j2 )
by A13, SEQM_3:41;

end;

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, Th13;

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 RLVECT_1:def 8

.= ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1))) by RLVECT_1:def 6 ;

then (1 / 2) * (G * (i1,j1)) = (1 / 2) * (G * (i1,(j1 + 1))) by Th3;

hence contradiction by A12, RLVECT_1:36; :: thesis: verum

end;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 RLVECT_1:def 8

.= ((1 / 2) * (G * (i1,j1))) + ((1 / 2) * (G * (i1,j1))) by RLVECT_1:def 6 ;

then (1 / 2) * (G * (i1,j1)) = (1 / 2) * (G * (i1,(j1 + 1))) by Th3;

hence contradiction by A12, RLVECT_1:36; :: thesis: verum

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, Th13;

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 RLVECT_1:def 8

.= ((1 / 2) * (G * (i1,j2))) + ((1 / 2) * (G * (i1,j2))) by RLVECT_1:def 6 ;

then (1 / 2) * (G * (i1,j1)) = (1 / 2) * (G * (i1,(j1 + 1))) by A15, Th3;

hence contradiction by A12, RLVECT_1:36; :: thesis: verum

end;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 RLVECT_1:def 8

.= ((1 / 2) * (G * (i1,j2))) + ((1 / 2) * (G * (i1,j2))) by RLVECT_1:def 6 ;

then (1 / 2) * (G * (i1,j1)) = (1 / 2) * (G * (i1,(j1 + 1))) by A15, Th3;

hence contradiction by A12, RLVECT_1:36; :: thesis: verum

hence j1 = j2 by Th2; :: thesis: verum