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
  r<>0 implies (r(#)f)"{0.V} = f"{0.V}
proof
  assume
A1: r<>0;
  now
    let c;
    thus c in (r(#)f)"{0.V} implies c in f"{0.V}
    proof
      assume
A2:   c in (r(#)f)"{0.V};
      then
A3:   c in dom (r(#)f) by PARTFUN2:26;
      (r(#)f)/.c in {0.V} by A2,PARTFUN2:26;
      then (r(#)f)/.c = 0.V by TARSKI:def 1;
      then r*f/.c = r*0.V by A3,Def4;
      then f/.c = 0.V by A1,RLVECT_1:36;
      then
A4:   f/.c in {0.V} by TARSKI:def 1;
      c in dom f by A3,Def4;
      hence thesis by A4,PARTFUN2:26;
    end;
    assume
A5: c in (f)"{0.V};
    then c in dom f by PARTFUN2:26;
    then
A6: c in dom (r(#)f) by Def4;
    f/.c in {0.V} by A5,PARTFUN2:26;
    then r*f/.c = r*0.V by TARSKI:def 1;
    then (r(#)f)/.c = 0.V by A6,Def4;
    then (r(#)f)/.c in {0.V} by TARSKI:def 1;
    hence c in (r(#)f)"{0.V} by A6,PARTFUN2:26;
  end;
  hence thesis by SUBSET_1:3;
end;
