let Rseq be Real_Sequence; :: thesis: for X being RealUnitarySpace
for seq1, seq2 being sequence of X holds Rseq * (seq1 + seq2) = (Rseq * seq1) + (Rseq * seq2)
let X be RealUnitarySpace; :: thesis: for seq1, seq2 being sequence of X holds Rseq * (seq1 + seq2) = (Rseq * seq1) + (Rseq * seq2)
let seq1, seq2 be sequence of X; :: thesis: Rseq * (seq1 + seq2) = (Rseq * seq1) + (Rseq * seq2)
let n be Element of NAT ; :: according to FUNCT_2:def 8 :: thesis: K13((Rseq * (seq1 + seq2)),n) = K13(((Rseq * seq1) + (Rseq * seq2)),n)
thus (Rseq * (seq1 + seq2)) . n = (Rseq . n) * ((seq1 + seq2) . n) by Def9
.= (Rseq . n) * ((seq1 . n) + (seq2 . n)) by NORMSP_1:def 2
.= ((Rseq . n) * (seq1 . n)) + ((Rseq . n) * (seq2 . n)) by RLVECT_1:def 5
.= ((Rseq * seq1) . n) + ((Rseq . n) * (seq2 . n)) by Def9
.= ((Rseq * seq1) . n) + ((Rseq * seq2) . n) by Def9
.= ((Rseq * seq1) + (Rseq * seq2)) . n by NORMSP_1:def 2 ; :: thesis: verum