reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;
reserve f for real-valued Function;
reserve f1,f2 for real-valued Function;
reserve f,f1,f2 for PartFunc of C,REAL;

theorem
  (f1|X is bounded_above & f2|Y is constant implies (f1-f2)|(X /\ Y) is
  bounded_above) & (f1|X is bounded_below & f2|Y is constant implies (f1-f2)|(X
/\ Y) is bounded_below) & (f1|X is bounded & f2|Y is constant implies (f1-f2)|(
X /\ Y) is bounded & (f2-f1)|(X /\ Y) is bounded& (f1(#)f2)|(X /\ Y) is bounded
  )
proof
  thus f1|X is bounded_above & f2|Y is constant implies (f1-f2)|(X /\ Y) is
  bounded_above
  proof
    assume that
A1: f1|X is bounded_above and
A2: f2|Y is constant;
    (-f2)|Y is constant by A2,Th90;
    hence thesis by A1,Th83;
  end;
  thus f1|X is bounded_below & f2|Y is constant implies (f1-f2)|(X /\ Y) is
  bounded_below
  proof
    assume that
A3: f1|X is bounded_below and
A4: f2|Y is constant;
    (-f2)|Y is constant by A4,Th90;
    hence thesis by A3,Th83;
  end;
  assume that
A5: f1|X is bounded and
A6: f2|Y is constant;
  (-f2)|Y is constant by A6,Th90;
  hence (f1-f2)|(X /\ Y) is bounded by A5,Th83;
  thus (f2-f1)|(X /\ Y) is bounded by A5,A6,Th84;
  thus thesis by A5,A6,Th84;
end;
