 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 Th19:
  for M be Matrix of n,m,F_Real st the_rank_of M = n
  for ME be Enumeration of (Mx2Tran M).:Affn st ME = (Mx2Tran M)*EN
  for pn st pn in Affin Affn holds pn |-- EN = ((Mx2Tran M).pn) |-- ME
 proof
  set TRn=TOP-REAL n;
  let M be Matrix of n,m,F_Real such that
   A1: the_rank_of M=n;
  set MT=Mx2Tran M;
  A2: MT is one-to-one by A1,MATRTOP1:39;
  set E=EN;
  set A=Affn;
  let ME be Enumeration of(Mx2Tran M).:A such that
   A3: ME=(Mx2Tran M)*E;
  dom MT=the carrier of TRn by FUNCT_2:def 1;
  then A,MT.:A are_equipotent by A2,CARD_1:33;
  then A4: card A=card(MT.:A) by CARD_1:5;
  let v be Element of TOP-REAL n such that
   A5: v in Affin A;
  set MTv=MT.v;
  A6: len(v|--E)=card A by Th16;
  A7: len(MTv|--ME)=card(MT.:A) by Th16;
  now let i be Nat;
   assume A8: 1<=i & i<=card A;
   then A9: i in dom(MTv|--ME) by A4,A7,FINSEQ_3:25;
   then A10: i in dom ME by FUNCT_1:11;
   A11: i in dom(v|--E) by A6,A8,FINSEQ_3:25;
   then i in dom E by FUNCT_1:11;
   then E.i in rng E by FUNCT_1:def 3;
   then reconsider Ei=E.i as Element of TRn;
   thus(v|--E).i=(v|--A).Ei by A11,FUNCT_1:12
    .=(MTv|--MT.:A).(MT.Ei) by A1,A5,MATRTOP2:25
    .=(MTv|--MT.:A).(ME.i) by A3,A10,FUNCT_1:12
    .=(MTv|--ME).i by A9,FUNCT_1:12;
  end;
  hence thesis by A4,A7,A6;
 end;
