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

theorem Th8:
  for x being UpperBound of A, y being UpperBound of B holds max(x,
  y) is UpperBound of A\/ B
proof
  let x be UpperBound of A, y be UpperBound of B;
  set m = max(x,y);
  let z;
  assume
A1: z in A \/ B;
  per cases by A1,XBOOLE_0:def 3;
  suppose
A2: z in A;
A3: x <= m by XXREAL_0:25;
    z <= x by A2,Def1;
    hence thesis by A3,XXREAL_0:2;
  end;
  suppose
A4: z in B;
A5: y <= m by XXREAL_0:25;
    z <= y by A4,Def1;
    hence thesis by A5,XXREAL_0:2;
  end;
end;
