reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem Th1:
for seq be Real_Sequence, h be PartFunc of REAL, the carrier of S
 st rng seq c= dom h holds seq.n in dom h
proof
   let seq be Real_Sequence;
   let h be PartFunc of REAL,the carrier of S;
   n in NAT by ORDINAL1:def 12; then
A1:n in dom seq by FUNCT_2:def 1;
   assume rng seq c= dom h; then
   n in dom ((h qua Function)*seq) by A1,RELAT_1:27;
   hence thesis by FUNCT_1:11;
end;
