
theorem Th5:
  for a,b being Real,s being Real_Sequence st rng s c= [.a,b
  .] holds s is sequence of Closed-Interval-MSpace(a,b)
proof
  let a,b be Real,s be Real_Sequence;
  assume
A1: rng s c= [.a,b.];
  reconsider t=s.0 as Real;
A2: dom s=NAT by FUNCT_2:def 1;
  then s.0 in rng s by FUNCT_1:def 3;
  then a<=t & t<=b by A1,XXREAL_1:1;
  then the carrier of Closed-Interval-MSpace(a,b)=[.a,b.] by TOPMETR:10
,XXREAL_0:2;
  hence thesis by A1,A2,FUNCT_2:2;
end;
