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;

theorem Th82:
  f|Y is bounded implies (abs f)|Y is bounded & (-f)|Y is bounded
proof
  assume
A1: f|Y is bounded;
  (abs f)|Y = abs(f|Y) by Th46;
  hence (abs f)|Y is bounded by A1;
  thus thesis by A1,Th80;
end;
