let x be ExtReal; :: thesis: sup {x} = x
A1: for z being UpperBound of {x} holds x <= z

proof

let z be UpperBound of {x}; :: thesis: x <= z
x in {x} by TARSKI:def 1;
hence x <= z by Def1; :: thesis: verum

end;

x is UpperBound of {x} by Th1;
hence sup {x} = x by A1, Def3; :: thesis: verum