reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;
reserve X,Y for set;

theorem Th53:
  f|Y is constant implies ||.f.|||Y is constant & (-f)|Y is constant
proof
  assume f|Y is constant;
  then consider r being VECTOR of V such that
A1: for c be Element of M st c in Y /\ dom f holds f/.c = r by PARTFUN2:35;
A2: ||.r.|| in REAL by XREAL_0:def 1;
  now
    let c be Element of M;
    assume
A3: c in Y /\ dom (||.f.||);
    then
A4: c in Y by XBOOLE_0:def 4;
A5: c in dom (||.f.||) by A3,XBOOLE_0:def 4;
    then c in dom f by NORMSP_0:def 3;
    then
A6: c in Y /\ dom f by A4,XBOOLE_0:def 4;
    thus (||.f.||).c = ||.f/.c.|| by A5,NORMSP_0:def 3
      .= ||.r.|| by A1,A6;
  end;
  hence ||.f.|||Y is constant by PARTFUN2:57,A2;
  now
    take p=-r;
    let c be Element of M;
    assume
A7: c in Y /\ dom (-f);
    then c in Y /\ dom f by VFUNCT_1:def 5;
    then
A8: -f/.c = p by A1;
    c in dom (-f) by A7,XBOOLE_0:def 4;
    hence (-f)/.c = p by A8,VFUNCT_1:def 5;
  end;
  hence thesis by PARTFUN2:35;
end;
