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;

theorem Th84:
  f1|X is bounded & f2|Y is bounded implies (f1(#)f2)|(X /\ Y) is
  bounded & (f1-f2)|(X /\ Y) is bounded
proof
  assume that
A1: f1|X is bounded and
A2: f2|Y is bounded;
  (f1(#)f2)|(X /\ Y) = f1|(X /\ Y)(#)f2|(X /\ Y) by Th45
    .= f1|(X /\ Y)(#)f2|Y|X by RELAT_1:71
    .= f1|X|Y(#)f2|Y|X by RELAT_1:71;
  hence (f1(#)f2)|(X /\ Y) is bounded by A1,A2;
  (-f2)|Y is bounded by A2,Th82;
  hence thesis by A1,Th83;
end;
