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 Th79:
  for A,B be finite Subset of V st
      A is affinely-independent & Affin A c= Affin B
    holds card A <= card B
 proof
  let A,B be finite Subset of V such that
   A1: A is affinely-independent and
   A2: Affin A c=Affin B;
  per cases;
  suppose A is empty;
   hence thesis;
  end;
  suppose A is non empty;
   then consider p be object such that
    A3: p in A;
   reconsider p as Element of V by A3;
   A4: A c=Affin A by Lm7;
   then A5: p in Affin A by A3;
   set LA=Lin(-p+A);
   A6: card A=card(-p+A) by Th7;
   {}V c=B;
   then consider Ib be affinely-independent Subset of V such that
    {}V c=Ib and
    A7: Ib c=B and
    A8: Affin Ib=Affin B by Th60;
   Ib is non empty by A2,A3,A8;
   then consider q be object such that
    A9: q in Ib;
   reconsider q as Element of V by A9;
   set LI=Lin(-q+Ib);
   A10: card Ib=card(-q+Ib) by Th7;
   -q+Ib c=the carrier of LI
   proof
    let x be object;
    assume x in -q+Ib;
    then x in LI by RLVECT_3:15;
    hence thesis;
   end;
   then reconsider qI=-q+Ib as finite Subset of LI by A7,A10;
   -q+q=0.V by RLVECT_1:5;
   then A11: 0.V in qI by A9;
   then A12: Lin(-q+Ib)=Lin(((-q+Ib)\{0.V})\/{0.V}) by ZFMISC_1:116
    .=Lin(-q+Ib\{0.V}) by Lm9
    .=Lin(qI\{0.V}) by RLVECT_5:20;
   A13: -q+Ib\{0.V} is linearly-independent by A9,Th41;
   then qI\{0.V} is linearly-independent by RLVECT_5:15;
   then qI\{0.V} is Basis of LI by A12,RLVECT_3:def 3;
   then reconsider LI as finite-dimensional Subspace of V by RLVECT_5:def 1;
   A14: Ib c=Affin Ib by Lm7;
   then A15: Affin Ib=q+Up LI by A9,Th57;
   A16: Affin A=p+Up LA by A3,A4,Th57;
   -p+A c=the carrier of LI
   proof
    let x be object;
    A17: 2+-1=1;
    2*q+(-1)*p=(1+1)*q+(-1)*p .=1*q+1*q+(-1)*p by RLVECT_1:def 6
     .=1*q+1*q+-p by RLVECT_1:16
     .=1*q+q+-p by RLVECT_1:def 8
     .=q+q+-p by RLVECT_1:def 8
     .=q+q-p by RLVECT_1:def 11
     .=(q-p)+q by RLVECT_1:28;
    then q+Up LI=q+LI & (q-p)+q in q+Up LI by A2,A8,A5,A9,A14,A15,A17,Th78,
RUSUB_4:30;
    then A18: q-p in LI by RLSUB_1:61;
    assume x in -p+A;
    then A19: x in LA by RLVECT_3:15;
    then x in V by RLSUB_1:9;
    then reconsider w=x as Element of V;
    w in Up LA by A19;
    then p+w in Affin A by A16;
    then p+w in q+Up LI by A2,A8,A15;
    then consider u be Element of V such that
     A20: p+w=q+u and
     A21: u in Up LI;
    A22: w=q+u-p by A20,RLVECT_4:1
     .=q+(u-p) by RLVECT_1:28
     .=u-(p-q) by RLVECT_1:29
     .=u+-(p-q) by RLVECT_1:def 11
     .=u+(q+-p) by RLVECT_1:33
     .=u+(q-p) by RLVECT_1:def 11;
    u in LI by A21;
    then w in LI by A18,A22,RLSUB_1:20;
    hence thesis;
   end;
   then reconsider LA as Subspace of LI by RLVECT_5:19;
   -p+A c=the carrier of LA
   proof
    let x be object;
    assume x in -p+A;
    then x in LA by RLVECT_3:15;
    hence thesis;
   end;
   then reconsider pA=-p+A as finite Subset of LA by A6;
   -p+p=0.V by RLVECT_1:5;
   then A23: 0.V in pA by A3;
   then A24: {0.V}c=pA by ZFMISC_1:31;
   A25: -p+A\{0.V} is linearly-independent by A1,A3,Th41;
   A26: card{0.V}=1 by CARD_1:30;
   A27: {0.V}c=qI by A11,ZFMISC_1:31;
   A28: dim LI=card(qI\{0.V}) by A13,A12,RLVECT_5:15,29
    .=card qI-1 by A27,A26,CARD_2:44;
   Lin(-p+A)=Lin(((-p+A)\{0.V})\/{0.V}) by A23,ZFMISC_1:116
    .=Lin(-p+A\{0.V}) by Lm9
    .=Lin(pA\{0.V}) by RLVECT_5:20;
   then dim LA=card(pA\{0.V}) by A25,RLVECT_5:15,29
    .=card A-1 by A6,A26,A24,CARD_2:44;
   then card A-1<=card qI-1 by A28,RLVECT_5:28;
   then A29: card A<=card qI by XREAL_1:9;
   card qI<=card B by A7,A10,NAT_1:43;
   hence thesis by A29,XXREAL_0:2;
  end;
 end;
