reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;
reserve I for affinely-independent Subset of V;

theorem
  for A,B be finite Subset of V st
      A is affinely-independent & Affin A = Affin B & card B <= card A
    holds B is affinely-independent
 proof
  let A,B be finite Subset of V;
  assume that
   A1: A is affinely-independent and
   A2: Affin A=Affin B and
   A3: card B<=card A;
  per cases;
  suppose A is empty;
   then B is empty by A3,XXREAL_0:1;
   hence thesis;
  end;
  suppose A is non empty;
   then consider p be object such that
    A4: p in A;
   card A>0 by A4;
   then reconsider n=(card A)-1 as Element of NAT by NAT_1:20;
   A5: A c=Affin A by Lm7;
   reconsider p as Element of V by A4;
   set L=Lin(-p+A);
   {}V c=B;
   then consider Ia be affinely-independent Subset of V such that
    {}V c=Ia and
    A6: Ia c=B and
    A7: Affin Ia=Affin B by Th60;
   Ia is non empty by A2,A4,A7;
   then consider q be object such that
    A8: q in Ia;
   -p+A c=[#]L
   proof
    let x be object;
    assume x in -p+A;
    then x in Lin(-p+A) by RLVECT_3:15;
    hence thesis;
   end;
   then reconsider pA=-p+A as Subset of L;
   -p+A\{0.V} is linearly-independent by A1,A4,Th41;
   then A9: pA\{0.V} is linearly-independent by RLVECT_5:15;
   -p+p=0.V by RLVECT_1:5;
   then A10: 0.V in pA by A4;
   then L=Lin(((-p+A)\{0.V})\/{0.V}) by ZFMISC_1:116
    .=Lin(-p+A\{0.V}) by Lm9
    .=Lin(pA\{0.V}) by RLVECT_5:20;
   then A11: pA\{0.V} is Basis of L by A9,RLVECT_3:def 3;
   reconsider IA=Ia as finite set by A6;
   A12: Ia c=Affin Ia by Lm7;
   reconsider q as Element of V by A8;
   p+L=p+Up L by RUSUB_4:30
    .=Affin A by A4,A5,Th57
    .=q+Up Lin(-q+Ia) by A2,A7,A8,A12,Th57
    .=q+Lin(-q+Ia) by RUSUB_4:30;
   then A13: L=Lin(-q+Ia) by RLSUB_1:69;
   set qI=-q+Ia;
   A14: qI c=[#]Lin(-q+Ia)
   proof
    let x be object;
    assume x in qI;
    then x in Lin(-q+Ia) by RLVECT_3:15;
    hence thesis;
   end;
   card pA=n+1 by Th7;
   then A15: card(pA\{0.V})=n by A10,STIRL2_1:55;
   then pA\{0.V} is finite;
   then A16: L is finite-dimensional by A11,RLVECT_5:def 1;
   reconsider qI as Subset of Lin(-q+Ia) by A14;
   -q+Ia\{0.V} is linearly-independent by A8,Th41;
   then A17: qI\{0.V} is linearly-independent by RLVECT_5:15;
   -q+q=0.V by RLVECT_1:5;
   then A18: 0.V in qI by A8;
   then Lin(-q+Ia)=Lin(((-q+Ia)\{0.V})\/{0.V}) by ZFMISC_1:116
    .=Lin(-q+Ia\{0.V}) by Lm9
    .=Lin(qI\{0.V}) by RLVECT_5:20;
   then qI\{0.V} is Basis of Lin(-q+Ia) by A17,RLVECT_3:def 3;
   then A19: card(qI\{0.V})=n by A11,A13,A15,A16,RLVECT_5:25;
   then (not 0.V in qI\{0.V}) & qI\{0.V} is finite by ZFMISC_1:56;
   then A20: n+1=card((qI\{0.V})\/{0.V}) by A19,CARD_2:41
    .=card qI by A18,ZFMISC_1:116
    .=card Ia by Th7;
   card IA<=card B by A6,NAT_1:43;
   hence thesis by A3,A6,A20,CARD_2:102,XXREAL_0:1;
  end;
 end;
