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
  |.f.||X is bounded_below
proof
  now
    take z=0;
    let c be object;
A1: z <= |. f/.c .| by COMPLEX1:46;
    assume c in X /\ dom (|.f.|);
    then
A2: c in dom (|.f.|) by XBOOLE_0:def 4;
    dom |.f.| = dom f by VALUED_1:def 11;
    then f.c = f/.c by A2,PARTFUN1:def 6;
    hence z <= (|.f.|).c by A1,VALUED_1:18;
  end;
  hence thesis by RFUNCT_1:71;
end;
