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

theorem Th10:
  sup(A \/ B) = max(sup A,sup B)
proof
  set m = max(sup A,sup B);
A1: sup B is UpperBound of B by Def3;
A2: for x being UpperBound of A \/ B holds m <= x
  proof
    let x be UpperBound of A \/ B;
    x is UpperBound of B by Th6,XBOOLE_1:7;
    then
A3: sup B <= x by Def3;
    x is UpperBound of A by Th6,XBOOLE_1:7;
    then sup A <= x by Def3;
    hence thesis by A3,XXREAL_0:28;
  end;
  sup A is UpperBound of A by Def3;
  then m is UpperBound of A \/ B by A1,Th8;
  hence thesis by A2,Def3;
end;
