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) )
assume i <= j ; :: thesis: len (Gauge D,i) <= len (Gauge D,j)
then A1: 2 |^ i <= 2 |^ j by PREPOWER:107;
( len (Gauge D,i) = (2 |^ i) + 3 & len (Gauge D,j) = (2 |^ j) + 3 ) by JORDAN8:def 1;
hence len (Gauge D,i) <= len (Gauge D,j) by A1, XREAL_1:8; :: thesis: verum