reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem Th54:
  f|Y is constant implies f is_bounded_on Y
proof
  assume f|Y is constant;
  then consider r being VECTOR of V such that
A1: for c st c in Y /\ dom f holds f/.c = r by PARTFUN2:35;
  now
    take p=||.r.||;
    let c;
    assume c in Y /\ dom f;
    hence ||.f/.c.|| <= p by A1;
  end;
  hence thesis;
end;
