 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 rE be Enumeration of r*Affv st v in Affin Affv & rE = r(#)EV & r<>0
    holds v|--EV = (r*v)|--rE
 proof
  set E=EV;
  let rE be Enumeration of r*Affv such that
   A1: v in Affin Affv and
   A2: rE=r(#)E and
   A3: r<>0;
  set vA=v|--Affv;
  A4: Carrier vA c=Affv by RLVECT_2:def 6;
  A5: r*v in {r*u:u in Affin Affv} by A1;
  A6: dom rE=dom E by A2,VFUNCT_1:def 4;
  Carrier(r(*)vA)=r*Carrier vA by A3,RLAFFIN1:23;
  then Carrier(r(*)vA)c=r*Affv by A4,CONVEX1:39;
  then reconsider rvA=r(*)vA as Linear_Combination of r*Affv by RLVECT_2:def 6;
  sum vA=1 by A1,RLAFFIN1:def 7;
  then A7: sum rvA=1 by A3,RLAFFIN1:38;
  Sum vA=v by A1,RLAFFIN1:def 7;
  then A8: Sum rvA=r*v by RLAFFIN1:40;
  A9: len((r*v)|--rE)=card(r*Affv) by Th16;
  A10: len(v|--E)=card Affv by Th16;
  rng E=Affv by Def1;
  then len E=card Affv by FINSEQ_4:62;
  then A11: dom(v|--E)=dom E by A10,FINSEQ_3:29;
  card Affv=card(r*Affv) by A3,Th12;
  then A12: dom(v|--E)=dom((r*v)|--rE) by A10,A9,FINSEQ_3:29;
  Affin(r*Affv)=r*Affin Affv by A3,RLAFFIN1:55;
  then r*v in Affin(r*Affv) by A5,CONVEX1:def 1;
  then A13: rvA=(r*v)|--(r*Affv) by A7,A8,RLAFFIN1:def 7;
  now let k be Nat;
   assume A14: k in dom(v|--E);
   then A15: (v|--E).k=vA.(E.k) & E/.k=E.k by A11,FUNCT_1:12,PARTFUN1:def 6;
   A16: rE/.k=r*(E/.k) by A2,A11,A6,A14,VFUNCT_1:def 4;
   ((r*v)|--rE).k=rvA.(rE.k) & rE/.k=rE.k by A13,A12,A11,A6,A14,FUNCT_1:12
,PARTFUN1:def 6;
   hence ((r*v)|--rE).k=vA.(r"*(r*(E/.k))) by A3,A16,RLAFFIN1:def 2
    .=vA.(r"*r*(E/.k)) by RLVECT_1:def 7
    .=vA.(1*(E/.k)) by A3,XCMPLX_0:def 7
    .=(v|--E).k by A15,RLVECT_1:def 8;
  end;
  hence thesis by A12,FINSEQ_1:13;
 end;
