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
  Y c= X & f|X is bounded implies f|Y is bounded
proof
  assume that
A1: Y c= X and
A2: f|X is bounded;
  |.f.||X = |.f|X.| by RFUNCT_1:46;
  then |.f.||X is bounded by A2,Lm3;
  then |.f.||Y = |.f|Y.| & |.f.||Y is bounded by A1,RFUNCT_1:46,74;
  hence thesis by Lm3;
end;
