 reserve x for set,
         n,m,k for Nat,
         r for Real,
         V for RealLinearSpace,
         v,u,w,t for VECTOR of V,
         Av for finite Subset of V,
         Affv for finite affinely-independent Subset of V;
reserve pn for Point of TOP-REAL n,
        An for Subset of TOP-REAL n,
        Affn for affinely-independent Subset of TOP-REAL n,
        Ak for Subset of TOP-REAL k;
reserve EV for Enumeration of Affv,
        EN for Enumeration of Affn;

theorem
  for E be Enumeration of Av holds
    r(#)E is Enumeration of r*Av
  iff
    r<>0 or Av is trivial
 proof
  let EV be Enumeration of Av;
  set rE=r(#)EV;
  A1: dom rE=dom EV by VFUNCT_1:def 4;
  then A2: len rE=len EV by FINSEQ_3:29;
  A3: rng EV=Av by Def1;
  then A4: card Av=len EV by FINSEQ_4:62;
  A5: rng rE=rE.:dom EV by A1,RELAT_1:113
   .=r*(EV.:dom EV) by Th10
   .=r*Av by A3,RELAT_1:113;
  A6: card{0.V}=1 by CARD_2:42;
  hereby reconsider rA=r*Av as finite set;
   assume rE is Enumeration of r*Av;
   then A7: card(r*Av)=card Av by A4,A2,A5,FINSEQ_4:62;
   assume r=0;
   then card Av<=1 by A6,A7,NAT_1:43,RLAFFIN1:12;
   then card Av<1+1 by NAT_1:13;
   hence Av is trivial by NAT_D:60;
  end;
  assume r<>0 or Av is trivial;
  then card Av=card(r*Av) by Th12;
  then rE is one-to-one by A4,A2,A5,FINSEQ_4:62;
  hence thesis by A5,Def1;
 end;
