 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;

theorem Th12:
  for A be Subset of V holds
    card A = card(r*A)
  iff
    r<>0 or A is trivial
proof
  let A be Subset of V;
  A1: card{0.V}=1 by CARD_2:42;
  hereby assume A2: card A=card(r*A);
   assume A3: r=0;
   then A4: r*A c={0.V} by RLAFFIN1:12;
   then reconsider a=A as finite set by A2;
   reconsider rA=r*A as finite set by A4;
   card(rA)<=card{0.V} by A3,NAT_1:43,RLAFFIN1:12;
   then card a<1+1 by A1,A2,NAT_1:13;
   hence A is trivial by NAT_D:60;
  end;
  assume A5: r<>0 or A is trivial;
  per cases by A5;
  suppose A6: r<>0;
   A7: 1*A=A & (1/r)*r*A=(1/r)*(r*A) by RLAFFIN1:10,11;
   A8: card(r*A)c=card A by Lm4;
   (1/r)*r=1 by A6,XCMPLX_1:87;
   then card A c=card(r*A) by A7,Lm4;
   hence thesis by A8;
  end;
  suppose A9: A is empty;
   r*A is empty
   proof
    assume r*A is non empty;
    then consider x be object such that
     A10: x in r*A;
    x in {r*v where v is Element of V:v in A} by A10,CONVEX1:def 1;
    then ex v be Element of V st x=r*v & v in A;
    hence contradiction by A9;
   end;
   hence thesis by A9;
  end;
  suppose A11: r=0 & A is 1-element;
   then consider x being object such that
    A12: A={x} by ZFMISC_1:131;
   r*A={0.V} by A11,CONVEX1:34;
   hence thesis by A1,A12,CARD_2:42;
  end;
end;
