theorem Th72:
  y <= x & x is LowerBound of A implies y is LowerBound of A
proof
  assume that
A1: y <= x and
A2: x is LowerBound of A;
  let z;
  assume z in A;
  then x <= z by A2,Def2;
  hence thesis by A1,XXREAL_0:2;
end;
