reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;
reserve A,B for ext-real-membered set;

theorem Th94:
  for A being non empty Subset of ExtREAL, r being ExtReal
  st r < sup A ex s being Element of ExtREAL st s in A & r < s
proof
  let A be non empty Subset of ExtREAL, r be ExtReal;
  assume
A1: r < sup A;
  assume
A2: for s being Element of ExtREAL st s in A holds not r < s;
  r is UpperBound of A
  proof
    let x be ExtReal;
    assume x in A;
    hence thesis by A2;
  end;
  hence contradiction by A1,Def3;
end;
