let a, b, c be real number ; :: thesis: ( a > 0 & a <> 1 & b > 0 & c > 0 implies (log a,b) + (log a,c) = log a,(b * c) )
assume that
A1: a > 0 and
A2: a <> 1 and
A3: b > 0 and
A4: c > 0 ; :: thesis: (log a,b) + (log a,c) = log a,(b * c)
A5: a to_power ((log a,b) + (log a,c)) = (a to_power (log a,b)) * (a to_power (log a,c)) by A1, Th32
.= b * (a to_power (log a,c)) by A1, A2, A3, Def3
.= b * c by A1, A2, A4, Def3 ;
b * c > 0 by A3, A4, XREAL_1:131;
hence (log a,b) + (log a,c) = log a,(b * c) by A1, A2, A5, Def3; :: thesis: verum