let i, j be Nat; 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); ( i <= j implies len (Gauge (D,i)) <= len (Gauge (D,j)) )
assume
i <= j
; 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; verum