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
  f is_bounded_on X & f is_bounded_on Y implies f is_bounded_on X \/ Y
proof
  assume that
A1: f is_bounded_on X and
A2: f is_bounded_on Y;
  consider r1 such that
A3: for c st c in X /\ dom f holds ||.f/.c.|| <= r1 by A1;
  consider r2 such that
A4: for c st c in Y /\ dom f holds ||.f/.c.|| <= r2 by A2;
  take r = |.r1.| + |.r2.|;
  let c;
  assume
A5: c in (X \/ Y) /\ dom f;
  then
A6: c in dom f by XBOOLE_0:def 4;
A7: c in X \/ Y by A5,XBOOLE_0:def 4;
  now
    per cases by A7,XBOOLE_0:def 3;
    suppose
      c in X;
      then c in X /\ dom f by A6,XBOOLE_0:def 4;
      then
A8:   ||.f/.c.|| <= r1 by A3;
A9:   0 <= |.r2.| by COMPLEX1:46;
      r1 <= |.r1.| by ABSVALUE:4;
      then ||.f/.c.|| <= |.r1.| by A8,XXREAL_0:2;
      then ||.f/.c.|| + 0 <= r by A9,XREAL_1:7;
      hence thesis;
    end;
    suppose
      c in Y;
      then c in Y /\ dom f by A6,XBOOLE_0:def 4;
      then
A10:  ||.f/.c.|| <= r2 by A4;
A11:  0 <= |.r1.| by COMPLEX1:46;
      r2 <= |.r2.| by ABSVALUE:4;
      then ||.f/.c.|| <= |.r2.| by A10,XXREAL_0:2;
      then 0 + ||.f/.c.|| <= r by A11,XREAL_1:7;
      hence thesis;
    end;
  end;
  hence thesis;
end;
