A1: p + r is FinSequence of REAL by Th19;
A2: len p = n by FINSEQ_1:def 18;
Seg (len (p + r)) = dom (p + r) by FINSEQ_1:def 3
.= dom p by VALUED_1:def 2
.= Seg n by A2, FINSEQ_1:def 3 ;
then len (p + r) = n by FINSEQ_1:8;
then p + r is Element of REAL n by A1, FINSEQ_2:110;
hence r + p is Point of (TOP-REAL n) by EUCLID:25; :: thesis: verum