let X be non empty set ; :: thesis: for a being Real
for f, g being Function of X, REAL
for F, G being Point of st f = F & g = G holds
( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
let a be Real; :: thesis: for f, g being Function of X, REAL
for F, G being Point of st f = F & g = G holds
( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
let f, g be Function of X, REAL ; :: thesis: for F, G being Point of st f = F & g = G holds
( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
let F, G be Point of ; :: thesis: ( f = F & g = G implies ( G = a * F iff for x being Element of X holds g . x = a * (f . x) ) )
reconsider f1 = F, g1 = G as VECTOR of ;
A1: ( G = a * F iff g1 = a * f1 ) ;
assume ( f = F & g = G ) ; :: thesis: ( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
hence ( G = a * F iff for x being Element of X holds g . x = a * (f . x) ) by A1, Th13; :: thesis: verum