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 Th86:
  f|X is bounded_below & f|Y is bounded_below implies f|(X \/ Y)
  is bounded_below
proof
  assume that
A1: f|X is bounded_below and
A2: f|Y is bounded_below;
  consider r1 such that
A3: for c being object st c in X /\ dom f holds r1 <= f.c by A1,Th71;
  consider r2 such that
A4: for c being object st c in Y /\ dom f holds r2 <= f.c by A2,Th71;
  now
    take r = -|.r1.| - |.r2.|;
    let c be object;
    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:     r1 <= f.c by A3;
A9:     0 <= |.r2.| by COMPLEX1:46;
        -|.r1.| <= r1 by ABSVALUE:4;
        then -|.r1.| <= f.c by A8,XXREAL_0:2;
        then r <= f.c - 0 by A9,XREAL_1:13;
        hence r <= f.c;
      end;
      suppose
        c in Y;
        then c in Y /\ dom f by A6,XBOOLE_0:def 4;
        then
A10:    r2 <= f.c by A4;
A11:    0 <= |.r1.| by COMPLEX1:46;
        -|.r2.| <= r2 by ABSVALUE:4;
        then -|.r2.| <= f.c by A10,XXREAL_0:2;
        then -|.r2.| - |.r1.| <= f.c - 0 by A11,XREAL_1:13;
        hence r <= f.c;
      end;
    end;
    hence r <= f.c;
  end;
  hence thesis by Th71;
end;
