let k, m be Nat; :: thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) st m > k holds
dist (((Gauge (C,m)) * (1,1)),((Gauge (C,m)) * (1,2))) < dist (((Gauge (C,k)) * (1,1)),((Gauge (C,k)) * (1,2)))
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: ( m > k implies dist (((Gauge (C,m)) * (1,1)),((Gauge (C,m)) * (1,2))) < dist (((Gauge (C,k)) * (1,1)),((Gauge (C,k)) * (1,2))) )
assume A1: m > k ; :: thesis: dist (((Gauge (C,m)) * (1,1)),((Gauge (C,m)) * (1,2))) < dist (((Gauge (C,k)) * (1,1)),((Gauge (C,k)) * (1,2)))
[1,(1 + 1)] in Indices (Gauge (C,k)) by Th6;
hence dist (((Gauge (C,m)) * (1,1)),((Gauge (C,m)) * (1,2))) < dist (((Gauge (C,k)) * (1,1)),((Gauge (C,k)) * (1,2))) by A1, Th5, Th8; :: thesis: verum