A1: p +* (x,r) is FinSequence of REAL by Th19;
A2: len p = n by CARD_1:def 7;
Seg (len (p +* (x,r))) = dom (p +* (x,r)) by FINSEQ_1:def 3
.= dom p by FUNCT_7:30
.= Seg n by A2, FINSEQ_1:def 3 ;
then len (p +* (x,r)) = n by FINSEQ_1:6;
then p +* (x,r) is Element of REAL n by A1, FINSEQ_2:92;
hence p +* (x,r) is Point of (TOP-REAL n) by EUCLID:22; :: thesis: verum