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 Th71:
  f|Y is bounded implies (r(#)f)|Y is bounded
proof
  assume
A1: f|Y is bounded;
  |.f.||Y = |.f|Y.| by RFUNCT_1:46;
  then |.f.||Y is bounded by A1,Lm3;
  then (|.r.|(#)|.f.|)|Y is bounded by RFUNCT_1:80;
  then |.r(#)f.||Y = |.(r(#)f)|Y.| & |.r(#)f.||Y is bounded by Th30,RFUNCT_1:46
;
  hence thesis by Lm3;
end;
