theorem Th71:
  x <= y & x is UpperBound of A implies y is UpperBound of A
proof
  assume that
A1: x <= y and
A2: x is UpperBound of A;
  let z;
  assume z in A;
  then z <= x by A2,Def1;
  hence thesis by A1,XXREAL_0:2;
end;
