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 Th75:
  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;
  consider r1 such that
A3: for c st c in X /\ dom f1 holds |.((f1/.c)).| <= r1 by A1,Th68;
  consider r2 such that
A4: for c st c in Y /\ dom f2 holds |.((f2/.c)).| <= r2 by A2,Th68;
  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 Th3;
    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;
    then |.((f1/.c)).|*|.((f2/.c)).| <= r by A10,A12,XREAL_1:66;
    then |.((f1/.c)) * ((f2/.c)).| <= r by COMPLEX1:65;
    hence |.(f1(#)f2)/.c.| <= r by A8,Th3;
  end;
  hence (f1(#)f2)|(X /\ Y) is bounded by Th68;
  (-f2)|Y is bounded by A2,Th73;
  hence thesis by A1,Th74;
end;
