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 Th52:
  f|Y is constant implies (p(#)f)|Y is constant
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
    let c;
    assume
A2: c in Y /\ dom (p(#)f);
    then
A3: c in Y by XBOOLE_0:def 4;
A4: c in dom (p(#)f) by A2,XBOOLE_0:def 4;
    then c in dom f by Def4;
    then
A5: c in Y /\ dom f by A3,XBOOLE_0:def 4;
    thus (p(#)f)/.c = p * f/.c by A4,Def4
      .= p*r by A1,A5;
  end;
  hence thesis by PARTFUN2:35;
end;
