let i, j be 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:93;
( 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_1:6; :: thesis: verum