reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;
reserve X,Y for set;

theorem
  f1 is_bounded_on X & f2|Y is constant implies f1-f2 is_bounded_on X /\
  Y & f2-f1 is_bounded_on X /\ Y
proof
  assume that
A1: f1 is_bounded_on X and
A2: f2|Y is constant;
A3: f2 is_bounded_on Y by A2,Th54;
  then -f2 is_bounded_on Y by Th45;
  then
A4: f1+-f2 is_bounded_on X /\ Y by A1,Th46;
  -f1 is_bounded_on X by A1,Th45;
  then f2+-f1 is_bounded_on Y /\ X by A3,Th46;
  hence thesis by A4,Th25;
end;
