let r be Real; :: thesis: for seq1, seq2 being Real_Sequence holds r (#) (seq1 + seq2) = (r (#) seq1) + (r (#) seq2)
let seq1, seq2 be Real_Sequence; :: thesis: r (#) (seq1 + seq2) = (r (#) seq1) + (r (#) seq2)

now :: thesis: for n being Element of NAT holds (r (#) (seq1 + seq2)) . n = ((r (#) seq1) + (r (#) seq2)) . n

let n be Element of NAT ; :: thesis: (r (#) (seq1 + seq2)) . n = ((r (#) seq1) + (r (#) seq2)) . n
thus (r (#) (seq1 + seq2)) . n = r * ((seq1 + seq2) . n) by Th9
.= r * ((seq1 . n) + (seq2 . n)) by Th7
.= (r * (seq1 . n)) + (r * (seq2 . n))
.= ((r (#) seq1) . n) + (r * (seq2 . n)) by Th9
.= ((r (#) seq1) . n) + ((r (#) seq2) . n) by Th9
.= ((r (#) seq1) + (r (#) seq2)) . n by Th7 ; :: thesis: verum

end;

hence r (#) (seq1 + seq2) = (r (#) seq1) + (r (#) seq2) by FUNCT_2:63; :: thesis: verum