reserve n,n1,m,k for Nat;
reserve x,y for set;
reserve s,g,g1,g2,r,p,p2,q,t for Real;
reserve s1,s2,s3 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve X for Subset of REAL;

theorem
  for X being open Subset of REAL, r st r in X ex g st 0<g & ].r-g,r+g.[ c= X
proof
  let X be open Subset of REAL, r;
  assume r in X;
  then consider N be Neighbourhood of r such that
A1: N c= X by Th18;
  consider g such that
A2: 0<g & N = ].r-g,r+g.[ by Def6;
  take g;
  thus thesis by A1,A2;
end;
