let a, c be positive heavy Real; :: thesis: for b, d being positive Real st log (a,b) < log (c,d) & a <= b holds
c < d
let b, d be positive Real; :: thesis: ( log (a,b) < log (c,d) & a <= b implies c < d )
assume A2: ( log (a,b) < log (c,d) & a <= b ) ; :: thesis: c < d
then log (a,b) >= 1 by AG1;
then log (c,d) > 1 by A2, XXREAL_0:2;
hence c < d by AG2; :: thesis: verum