let S be RealNormSpace; :: thesis: for seq being sequence of S holds - seq = (- 1) * seq
let seq be sequence of S; :: thesis: - seq = (- 1) * seq

now :: thesis: for n being Element of NAT holds ((- 1) * seq) . n = (- seq) . n

let n be Element of NAT ; :: thesis: ((- 1) * seq) . n = (- seq) . n
thus ((- 1) * seq) . n = (- 1) * (seq . n) by NORMSP_1:def 5
.= - (seq . n) by RLVECT_1:16
.= (- seq) . n by BHSP_1:44 ; :: thesis: verum

end;

hence - seq = (- 1) * seq by FUNCT_2:63; :: thesis: verum