let G be Go-board; :: thesis: for i1, j1, i2, j2 being Nat st 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) holds
( ( i1 = i2 or i1 = i2 + 1 ) & ( j1 = j2 or j1 + 1 = j2 ) )

let i1, j1, i2, j2 be Nat; :: thesis: ( 1 <= i1 & i1 <= len G & 1 <= j1 & j1 + 1 <= width G & 1 <= i2 & i2 + 1 <= len G & 1 <= j2 & j2 <= width G & LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) implies ( ( i1 = i2 or i1 = i2 + 1 ) & ( j1 = j2 or j1 + 1 = j2 ) ) )
assume that
A1: 1 <= i1 and
A2: i1 <= len G and
A3: 1 <= j1 and
A4: j1 + 1 <= width G and
A5: 1 <= i2 and
A6: i2 + 1 <= len G and
A7: 1 <= j2 and
A8: j2 <= width G ; :: thesis: ( not LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) or ( ( i1 = i2 or i1 = i2 + 1 ) & ( j1 = j2 or j1 + 1 = j2 ) ) )
assume LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) meets LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) ; :: thesis: ( ( i1 = i2 or i1 = i2 + 1 ) & ( j1 = j2 or j1 + 1 = j2 ) )
then consider x being object such that
A9: x in LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) and
A10: x in LSeg ((G * (i2,j2)),(G * ((i2 + 1),j2))) by XBOOLE_0:3;
reconsider p = x as Point of () by A9;
consider r1 being Real such that
A11: p = ((1 - r1) * (G * (i1,j1))) + (r1 * (G * (i1,(j1 + 1)))) and
A12: r1 >= 0 and
A13: r1 <= 1 by A9;
consider r2 being Real such that
A14: p = ((1 - r2) * (G * (i2,j2))) + (r2 * (G * ((i2 + 1),j2))) and
A15: r2 >= 0 and
A16: r2 <= 1 by A10;
A17: i2 < len G by ;
A18: 1 <= j1 + 1 by ;
then A19: (G * (i1,(j1 + 1))) `2 = (G * (1,(j1 + 1))) `2 by
.= (G * (i2,(j1 + 1))) `2 by ;
A20: j1 < width G by ;
then A21: (G * (i1,j1)) `2 = (G * (1,j1)) `2 by
.= (G * (i2,j1)) `2 by ;
A22: ((1 - r2) * ((G * (i2,j2)) `2)) + (r2 * ((G * ((i2 + 1),j2)) `2)) = (((1 - r2) * (G * (i2,j2))) `2) + (r2 * ((G * ((i2 + 1),j2)) `2)) by TOPREAL3:4
.= (((1 - r2) * (G * (i2,j2))) `2) + ((r2 * (G * ((i2 + 1),j2))) `2) by TOPREAL3:4
.= p `2 by
.= (((1 - r1) * (G * (i1,j1))) `2) + ((r1 * (G * (i1,(j1 + 1)))) `2) by
.= ((1 - r1) * ((G * (i1,j1)) `2)) + ((r1 * (G * (i1,(j1 + 1)))) `2) by TOPREAL3:4
.= ((1 - r1) * ((G * (i2,j1)) `2)) + (r1 * ((G * (i2,(j1 + 1))) `2)) by ;
A23: 1 <= i2 + 1 by ;
thus ( i1 = i2 or i1 = i2 + 1 ) :: thesis: ( j1 = j2 or j1 + 1 = j2 )
proof
A24: (G * (i2,j2)) `1 = (G * (i2,1)) `1 by
.= (G * (i2,j1)) `1 by ;
A25: (G * ((i2 + 1),j2)) `1 = (G * ((i2 + 1),1)) `1 by
.= (G * ((i2 + 1),j1)) `1 by ;
A26: ((1 - r1) * ((G * (i1,j1)) `1)) + (r1 * ((G * (i1,(j1 + 1))) `1)) = (((1 - r1) * (G * (i1,j1))) `1) + (r1 * ((G * (i1,(j1 + 1))) `1)) by TOPREAL3:4
.= (((1 - r1) * (G * (i1,j1))) `1) + ((r1 * (G * (i1,(j1 + 1)))) `1) by TOPREAL3:4
.= p `1 by
.= (((1 - r2) * (G * (i2,j2))) `1) + ((r2 * (G * ((i2 + 1),j2))) `1) by
.= ((1 - r2) * ((G * (i2,j2)) `1)) + ((r2 * (G * ((i2 + 1),j2))) `1) by TOPREAL3:4
.= ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) by ;
A27: (G * (i1,j1)) `1 = (G * (i1,1)) `1 by
.= (G * (i1,(j1 + 1))) `1 by ;
assume A28: ( not i1 = i2 & not i1 = i2 + 1 ) ; :: thesis: contradiction
per cases ( ( i1 < i2 & i1 < i2 + 1 ) or ( i1 < i2 & i2 + 1 < i1 ) or ( i2 < i1 & i1 < i2 + 1 ) or i2 + 1 < i1 ) by ;
suppose A29: ( i1 < i2 & i1 < i2 + 1 ) ; :: thesis: contradiction
i2 < i2 + 1 by XREAL_1:29;
then (G * (i2,j1)) `1 < (G * ((i2 + 1),j1)) `1 by ;
then A30: ( ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * (i2,j1)) `1)) = 1 * ((G * (i2,j1)) `1) & r2 * ((G * (i2,j1)) `1) <= r2 * ((G * ((i2 + 1),j1)) `1) ) by ;
(G * (i1,j1)) `1 < (G * (i2,j1)) `1 by ;
hence contradiction by A26, A27, A30, XREAL_1:6; :: thesis: verum
end;
suppose ( i1 < i2 & i2 + 1 < i1 ) ; :: thesis: contradiction
end;
suppose ( i2 < i1 & i1 < i2 + 1 ) ; :: thesis: contradiction
end;
suppose A31: i2 + 1 < i1 ; :: thesis: contradiction
i2 < i2 + 1 by XREAL_1:29;
then A32: (G * (i2,j1)) `1 <= (G * ((i2 + 1),j1)) `1 by ;
1 - r2 >= 0 by ;
then ( ((1 - r2) * ((G * ((i2 + 1),j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) = 1 * ((G * ((i2 + 1),j1)) `1) & (1 - r2) * ((G * (i2,j1)) `1) <= (1 - r2) * ((G * ((i2 + 1),j1)) `1) ) by ;
then ((1 - r2) * ((G * (i2,j1)) `1)) + (r2 * ((G * ((i2 + 1),j1)) `1)) <= (G * ((i2 + 1),j1)) `1 by XREAL_1:6;
hence contradiction by A2, A3, A20, A23, A26, A27, A31, GOBOARD5:3; :: thesis: verum
end;
end;
end;
A33: (G * (i2,j2)) `2 = (G * (1,j2)) `2 by
.= (G * ((i2 + 1),j2)) `2 by ;
assume A34: ( not j1 = j2 & not j1 + 1 = j2 ) ; :: thesis: contradiction
per cases ( ( j2 < j1 & j2 < j1 + 1 ) or ( j2 < j1 & j1 + 1 < j2 ) or ( j1 < j2 & j2 < j1 + 1 ) or j1 + 1 < j2 ) by ;
suppose A35: ( j2 < j1 & j2 < j1 + 1 ) ; :: thesis: contradiction
j1 < j1 + 1 by XREAL_1:29;
then (G * (i2,j1)) `2 < (G * (i2,(j1 + 1))) `2 by ;
then A36: ( ((1 - r1) * ((G * (i2,j1)) `2)) + (r1 * ((G * (i2,j1)) `2)) = 1 * ((G * (i2,j1)) `2) & r1 * ((G * (i2,j1)) `2) <= r1 * ((G * (i2,(j1 + 1))) `2) ) by ;
(G * (i2,j2)) `2 < (G * (i2,j1)) `2 by ;
hence contradiction by A22, A33, A36, XREAL_1:6; :: thesis: verum
end;
suppose ( j2 < j1 & j1 + 1 < j2 ) ; :: thesis: contradiction
end;
suppose ( j1 < j2 & j2 < j1 + 1 ) ; :: thesis: contradiction
end;
suppose A37: j1 + 1 < j2 ; :: thesis: contradiction
j1 < j1 + 1 by XREAL_1:29;
then A38: (G * (i2,j1)) `2 <= (G * (i2,(j1 + 1))) `2 by ;
1 - r1 >= 0 by ;
then ( ((1 - r1) * ((G * (i2,(j1 + 1))) `2)) + (r1 * ((G * (i2,(j1 + 1))) `2)) = 1 * ((G * (i2,(j1 + 1))) `2) & (1 - r1) * ((G * (i2,j1)) `2) <= (1 - r1) * ((G * (i2,(j1 + 1))) `2) ) by ;
then ((1 - r1) * ((G * (i2,j1)) `2)) + (r1 * ((G * (i2,(j1 + 1))) `2)) <= (G * (i2,(j1 + 1))) `2 by XREAL_1:6;
hence contradiction by A5, A8, A18, A17, A22, A33, A37, GOBOARD5:4; :: thesis: verum
end;
end;