let i, j be Element of NAT ; :: thesis: for D being non empty Subset of (TOP-REAL 2) st i < j holds
len (Gauge D,i) < len (Gauge D,j)
let D be non empty Subset of (TOP-REAL 2); :: thesis: ( i < j implies len (Gauge D,i) < len (Gauge D,j) )
A1: ( len (Gauge D,i) = (2 |^ i) + 3 & len (Gauge D,j) = (2 |^ j) + 3 ) by JORDAN8:def 1;
assume i < j ; :: thesis: len (Gauge D,i) < len (Gauge D,j)
then 2 |^ i < 2 |^ j by PEPIN:71;
hence len (Gauge D,i) < len (Gauge D,j) by A1, XREAL_1:8; :: thesis: verum