let f be non empty FinSequence of (); :: thesis: for I, i1, i being Nat
for Y being non empty finite Subset of NAT st 1 <= i & i <= len f & 1 <= I & I <= len (GoB f) & Y = { j where j is Element of NAT : ( [I,j] in Indices (GoB f) & ex k being Nat st
( k in dom f & f /. k = (GoB f) * (I,j) ) )
}
& (f /. i) `1 = ((GoB f) * (I,1)) `1 & i1 = min Y holds
((GoB f) * (I,i1)) `2 <= (f /. i) `2

let I, i1, i be Nat; :: thesis: for Y being non empty finite Subset of NAT st 1 <= i & i <= len f & 1 <= I & I <= len (GoB f) & Y = { j where j is Element of NAT : ( [I,j] in Indices (GoB f) & ex k being Nat st
( k in dom f & f /. k = (GoB f) * (I,j) ) )
}
& (f /. i) `1 = ((GoB f) * (I,1)) `1 & i1 = min Y holds
((GoB f) * (I,i1)) `2 <= (f /. i) `2

let Y be non empty finite Subset of NAT; :: thesis: ( 1 <= i & i <= len f & 1 <= I & I <= len (GoB f) & Y = { j where j is Element of NAT : ( [I,j] in Indices (GoB f) & ex k being Nat st
( k in dom f & f /. k = (GoB f) * (I,j) ) )
}
& (f /. i) `1 = ((GoB f) * (I,1)) `1 & i1 = min Y implies ((GoB f) * (I,i1)) `2 <= (f /. i) `2 )

A1: f /. i = |[((f /. i) `1),((f /. i) `2)]| by EUCLID:53;
assume A2: ( 1 <= i & i <= len f & 1 <= I & I <= len (GoB f) & Y = { j where j is Element of NAT : ( [I,j] in Indices (GoB f) & ex k being Nat st
( k in dom f & f /. k = (GoB f) * (I,j) ) )
}
& (f /. i) `1 = ((GoB f) * (I,1)) `1 & i1 = min Y ) ; :: thesis: ((GoB f) * (I,i1)) `2 <= (f /. i) `2
then A3: i in dom f by FINSEQ_3:25;
then consider i2, j2 being Nat such that
A4: [i2,j2] in Indices (GoB f) and
A5: f /. i = (GoB f) * (i2,j2) by GOBOARD5:11;
A6: j2 <= width (GoB f) by ;
A7: 1 <= j2 by ;
then A8: [I,j2] in Indices (GoB f) by ;
A9: i2 <= len (GoB f) by ;
1 <= i2 by ;
then A10: (f /. i) `2 = ((GoB f) * (1,j2)) `2 by
.= ((GoB f) * (I,j2)) `2 by ;
i1 in Y by ;
then ex j being Element of NAT st
( i1 = j & [I,j] in Indices (GoB f) & ex k being Nat st
( k in dom f & f /. k = (GoB f) * (I,j) ) ) by A2;
then A11: 1 <= i1 by MATRIX_0:32;
A12: j2 in NAT by ORDINAL1:def 12;
(f /. i) `1 = ((GoB f) * (I,j2)) `1 by ;
then f /. i = (GoB f) * (I,j2) by ;
then j2 in Y by A2, A3, A8, A12;
then A13: i1 <= j2 by ;
A14: j2 <= width (GoB f) by ;
now :: thesis: ( ( i1 < j2 & ((GoB f) * (I,i1)) `2 <= ((GoB f) * (I,j2)) `2 ) or ( i1 >= j2 & ((GoB f) * (I,i1)) `2 <= ((GoB f) * (I,j2)) `2 ) )
per cases ( i1 < j2 or i1 >= j2 ) ;
case i1 < j2 ; :: thesis: ((GoB f) * (I,i1)) `2 <= ((GoB f) * (I,j2)) `2
hence ((GoB f) * (I,i1)) `2 <= ((GoB f) * (I,j2)) `2 by ; :: thesis: verum
end;
case i1 >= j2 ; :: thesis: ((GoB f) * (I,i1)) `2 <= ((GoB f) * (I,j2)) `2
hence ((GoB f) * (I,i1)) `2 <= ((GoB f) * (I,j2)) `2 by ; :: thesis: verum
end;
end;
end;
hence ((GoB f) * (I,i1)) `2 <= (f /. i) `2 by A10; :: thesis: verum