reserve n,k,k1,m,m1,n1,n2,l for Nat;
reserve r,r1,r2,p,p1,g,g1,g2,s,s1,s2,t for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve Nseq for increasing sequence of NAT;
reserve x for set;
reserve X,Y for Subset of REAL;
reserve k,n for Nat,
  r,r9,r1,r2 for Real,
  c,c9,c1,c2,c3 for Element of COMPLEX;
reserve z,z1,z2 for FinSequence of COMPLEX;
reserve x,z,z1,z2,z3 for Element of COMPLEX n,
  A,B for Subset of COMPLEX n;
reserve
  v,v1,v2 for FinSequence of REAL,
  n,m,k for Nat,
  x for set;

theorem
  for R being non empty Subset of REAL,r0 being Real st for
  r being Real st r in R holds r <= r0 holds upper_bound R <= r0
proof
  let R be non empty Subset of REAL,r0 be Real;
  assume
A1: for r being Real st r in R holds r<=r0;
  then for r being ExtReal st r in R holds r<=r0;
  then r0 is UpperBound of R by XXREAL_2:def 1;
  then
A2: R is bounded_above;
  now
    set r1=(upper_bound R) -r0;
    assume upper_bound R >r0;
    then r1>0 by XREAL_1:50;
    then ex r being Real st r in R & (upper_bound R)-r1<r
     by A2,Def1;
    hence contradiction by A1;
  end;
  hence thesis;
end;
