reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem
  f|X is bounded & f|Y is bounded implies f|(X \/ Y) is bounded
proof
  assume that
A1: f|X is bounded and
A2: f|Y is bounded;
  |.f.||Y = |.f|Y.| by RFUNCT_1:46;
  then
A3: |.f.||Y is bounded by A2,Lm3;
  |.f.||X = |.f|X.| by RFUNCT_1:46;
  then |.f.||X is bounded by A1,Lm3;
  then |.f.||(X \/ Y) = |.f|(X \/ Y).| & |.f.||(X \/ Y) is bounded by A3,
RFUNCT_1:46,87;
  hence thesis by Lm3;
end;
