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;
