 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 Th15:
  for M be Matrix of n,m,F_Real st the_rank_of M = n
  for A be finite Subset of TOP-REAL n, E be Enumeration of A
    holds (Mx2Tran M)*E is Enumeration of(Mx2Tran M).:A
 proof
  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 TRn=TOP-REAL n;
  let A be finite Subset of TOP-REAL n;
  let E be Enumeration of A;
  A3: rng E=A by Def1;
  dom MT=[#]TRn by FUNCT_2:def 1;
  then len(MT*E)=len E by A3,FINSEQ_2:29;
  then A4: dom(MT*E)=dom E by FINSEQ_3:29;
  rng(MT*E)=(MT*E).:dom(MT*E) by RELAT_1:113
   .=MT.:(E.:dom E) by A4,RELAT_1:126
   .=MT.:A by A3,RELAT_1:113;
  hence thesis by A2,Def1;
 end;
