 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 Th17:
  for E be Enumeration of v+Affv st w in Affin Affv & E = EV+(card Affv|->v)
    holds w|--EV=(v+w)|--E
proof
  set E=EV;
  let Ev be Enumeration of v+Affv such that
   A1: w in Affin Affv and
   A2: Ev=E+(card Affv|->v);
  set wA=w|--Affv;
  A3: sum wA=1 by A1,RLAFFIN1:def 7;
  v+w in {v+u:u in Affin Affv} by A1;
  then A4: v+w in v+Affin Affv by RUSUB_4:def 8;
  rng E=Affv by Def1;
  then A5: len E=card Affv by FINSEQ_4:62;
  then reconsider e=E,cAv=card Affv|->v as Element of card Affv-tuples_on the
carrier of V by FINSEQ_2:92;
  A6: Affin(v+Affv)=v+Affin Affv & 1*v=v by RLAFFIN1:53,RLVECT_1:def 8;
  Carrier(v+wA)=v+Carrier wA & Carrier wA c=Affv by RLAFFIN1:16,RLVECT_2:def 6;
  then Carrier(v+wA)c=v+Affv by RLTOPSP1:8;
  then reconsider vwA=v+wA as Linear_Combination of v+Affv by RLVECT_2:def 6;
  Sum wA=w by A1,RLAFFIN1:def 7;
  then A7: Sum vwA=1*v+w by A3,RLAFFIN1:39;
  A8: len(w|--E)=card Affv by Th16;
  A9: card Affv=card(v+Affv) by RLAFFIN1:7;
  then len((v+w)|--Ev)=card Affv by Th16;
  then A10: dom(w|--E)=dom((v+w)|--Ev) by A8,FINSEQ_3:29;
  rng Ev=v+Affv by Def1;
  then A11: len Ev=card Affv by A9,FINSEQ_4:62;
  sum vwA=1 by A3,RLAFFIN1:37;
  then A12: vwA=(v+w)|--(v+Affv) by A4,A7,A6,RLAFFIN1:def 7;
  now let i be Nat;
   assume A13: i in dom(w|--E);
   then A14: (w|--E).i=(w|--Affv).(E.i) by FUNCT_1:12;
   dom E=dom(w|--E) by A8,A5,FINSEQ_3:29;
   then A15: E.i=E/.i by A13,PARTFUN1:def 6;
   i in Seg card Affv by A8,A13,FINSEQ_1:def 3;
   then A16: cAv.i=v by FINSEQ_2:57;
   A17: ((v+w)|--Ev).i=((v+w)|--(v+Affv)).(Ev.i) by A10,A13,FUNCT_1:12;
   dom Ev=dom(w|--E) by A8,A11,FINSEQ_3:29;
   then Ev.i=E/.i+v by A2,A13,A16,A15,FVSUM_1:17;
   hence ((v+w)|--Ev).i=(w|--Affv).(E/.i+v-v) by A12,A17,RLAFFIN1:def 1
    .=(w|--Affv).(E/.i+(v-v)) by RLVECT_1:28
    .=(w|--Affv).(E/.i+0.V) by RLVECT_1:15
    .=(w|--E).i by A14,A15,RLVECT_1:def 4;
  end;
  hence thesis by A10,FINSEQ_1:13;
 end;
