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

theorem Th29:
  x <= y implies sup [.x,y.] = y
proof
  assume
A1: x <= y;
A2: for z being UpperBound of [.x,y.] holds y <= z
  proof
    let z be UpperBound of [.x,y.];
    y in [.x,y.] by A1,XXREAL_1:1;
    hence thesis by Def1;
  end;
  y is UpperBound of [.x,y.] by Th21;
  hence thesis by A2,Def3;
end;
