reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem
  for f1 be PartFunc of C,REAL holds f1|X is bounded & f2 is_bounded_on
  Y implies f1(#)f2 is_bounded_on (X /\ Y)
proof
  let f1 be PartFunc of C,REAL;
  assume that
A1: f1|X is bounded and
A2: f2 is_bounded_on Y;
  consider r1 be Real such that
A3: for c being object st c in X /\ dom f1 holds |.f1.c.| <= r1
     by A1,RFUNCT_1:73;
  consider r2 such that
A4: for c st c in Y /\ dom f2 holds ||.(f2/.c).|| <= r2 by A2;
  reconsider r1 as Real;
  now
    take r=r1*r2;
    let c;
    assume
A5: c in X /\ Y /\ dom (f1(#)f2);
    then
A6: c in X /\ Y by XBOOLE_0:def 4;
    then
A7: c in X by XBOOLE_0:def 4;
A8: c in dom (f1(#)f2) by A5,XBOOLE_0:def 4;
    then
A9: c in dom f1 /\ dom f2 by Def3;
    then c in dom f1 by XBOOLE_0:def 4;
    then c in X /\ dom f1 by A7,XBOOLE_0:def 4;
    then
A10: |.f1.c.| <= r1 by A3;
A11: c in Y by A6,XBOOLE_0:def 4;
    c in dom f2 by A9,XBOOLE_0:def 4;
    then c in Y /\ dom f2 by A11,XBOOLE_0:def 4;
    then
A12: ||.(f2/.c).|| <= r2 by A4;
    0<=|.f1.c.| & 0<=||.(f2/.c).|| by COMPLEX1:46,NORMSP_1:4;
    then |.f1.c.|*||.(f2/.c).|| <= r by A10,A12,XREAL_1:66;
    then ||.f1.c * (f2/.c).|| <= r by NORMSP_1:def 1;
    hence ||.(f1(#)f2)/.c.|| <= r by A8,Def3;
  end;
  hence thesis;
end;
