let seq be Real_Sequence; :: thesis: ( seq is bounded_above implies (superior_realsequence seq) . 0 = sup seq )
reconsider Y1 = { (seq . k) where k is Element of NAT : 0 <= k } as Subset of REAL by Th29;
(superior_realsequence seq) . 0 = sup Y1 by Def5
.= sup seq by SETLIM_1:5 ;
hence ( seq is bounded_above implies (superior_realsequence seq) . 0 = sup seq ) ; :: thesis: verum