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 Th74:
  f1|X is bounded & f2|Y is bounded implies (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;
  ex p1 st for c st c in X /\ Y /\ dom (f1+f2) holds |.(f1+f2)/.c.| <= p1
  proof
    take r0=r1+r2;
    let c;
A5: |.((f1/.c)) + ((f2/.c)).| <= |.((f1/.c)).|+|.((f2/.c)).| by COMPLEX1:56;
    assume
A6: c in X /\ Y /\ dom (f1+f2);
    then
A7: c in X /\ Y by XBOOLE_0:def 4;
    then
A8: c in X by XBOOLE_0:def 4;
A9: c in Y by A7,XBOOLE_0:def 4;
A10: c in dom (f1+f2) by A6,XBOOLE_0:def 4;
    then
A11: c in dom f1 /\ dom f2 by VALUED_1:def 1;
    then c in dom f2 by XBOOLE_0:def 4;
    then c in Y /\ dom f2 by A9,XBOOLE_0:def 4;
    then
A12: |.((f2/.c)).| <= r2 by A4;
    c in dom f1 by A11,XBOOLE_0:def 4;
    then c in X /\ dom f1 by A8,XBOOLE_0:def 4;
    then |.((f1/.c)).| <= r1 by A3;
    then |.((f1/.c)).|+|.((f2/.c)).| <= r0 by A12,XREAL_1:7;
    then |.((f1/.c)) + ((f2/.c)).| <= r0 by A5,XXREAL_0:2;
    hence thesis by A10,Th1;
  end;
  hence thesis by Th68;
end;
