let a, b, c be real number ; :: thesis: ( a > 0 & a <> 1 & b > 0 & b <> 1 & c > 0 implies log a,c = (log a,b) * (log b,c) )
assume A1: ( a > 0 & a <> 1 & b > 0 & b <> 1 & c > 0 ) ; :: thesis: log a,c = (log a,b) * (log b,c)
then a to_power ((log a,b) * (log b,c)) = (a to_power (log a,b)) to_power (log b,c) by Th38
.= b to_power (log b,c) by A1, Def3
.= c by A1, Def3 ;
hence log a,c = (log a,b) * (log b,c) by A1, Def3; :: thesis: verum