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 Th59:
  Affin A = {Sum L where L is Linear_Combination of A : sum L=1}
 proof
  set S={Sum L where L is Linear_Combination of A:sum L=1};
  per cases;
  suppose A1: A is empty;
   assume Affin A<>S;
   then S is non empty by A1;
   then consider x being object such that
    A2: x in S;
   consider L be Linear_Combination of A such that
    x=Sum L and
    A3: sum L=1 by A2;
   A={}(the carrier of V) by A1;
   then L=ZeroLC(V) by RLVECT_2:23;
   hence thesis by A3,Th31;
  end;
  suppose A is non empty;
   then consider p be object such that
    A4: p in A;
   reconsider p as Element of V by A4;
   A c=Affin A by Lm7;
   then A5: Affin A=p+Up Lin(-p+A) by A4,Th57;
   thus Affin A c=S
   proof
    let x be object;
    assume x in Affin A;
    then consider v such that
     A6: x=p+v and
     A7: v in Up Lin(-p+A) by A5;
    v in Lin(-p+A) by A7;
    then consider L be Linear_Combination of-p+A such that
     A8: Sum L=v by RLVECT_3:14;
    set pL=p+L;
    consider Lp be Linear_Combination of{0.V} such that
     A9: Lp.0.V=1-sum L by RLVECT_4:37;
    set pLL=p+(L+Lp);
    set pLp=p+Lp;
    A10: Carrier Lp c={0.V} by RLVECT_2:def 6;
    then A11: p+Carrier Lp c=p+{0.V} by RLTOPSP1:8;
    A12: Carrier pL=p+Carrier L & Carrier L c=-p+A by Th16,RLVECT_2:def 6;
    p+(-p+A)=(p+-p)+A by Th5
     .=0.V+A by RLVECT_1:5
     .=A by Th6;
    then A13: Carrier pL c=A by A12,RLTOPSP1:8;
    A14: Carrier(pL+pLp)c=Carrier pL\/Carrier pLp & pLL=pL+pLp by Th17,
RLVECT_2:37;
    Carrier pLp=p+Carrier Lp & p+{0.V}={p+0.V} by Lm3,Th16;
    then Carrier pLp c={p} by A11;
    then Carrier pL\/Carrier pLp c=A\/{p} by A13,XBOOLE_1:13;
    then Carrier pLL c=A\/{p} by A14;
    then Carrier pLL c=A by A4,ZFMISC_1:40;
    then A15: pLL is Linear_Combination of A by RLVECT_2:def 6;
    A16: sum pLL=sum(L+Lp) by Th37;
    A17: sum(L+Lp)=sum L+sum Lp by Th34
     .=sum L+(1-sum L) by A9,A10,Th32
     .=1;
    then Sum pLL=1*p+Sum(L+Lp) by Th39
     .=1*p+(v+Sum Lp) by A8,RLVECT_3:1
     .=1*p+(v+Lp.0.V*0.V) by RLVECT_2:32
     .=1*p+(v+0.V)
     .=p+(v+0.V) by RLVECT_1:def 8
     .=x by A6;
    hence thesis by A15,A16,A17;
   end;
   let x be object;
   assume x in S;
   then consider L be Linear_Combination of A such that
    A18: Sum L=x and
    A19: sum L=1;
   set pL=-p+L;
   Carrier L c=A by RLVECT_2:def 6;
   then A20: -p+Carrier L c=-p+A by RLTOPSP1:8;
   -p+Carrier L=Carrier pL by Th16;
   then pL is Linear_Combination of-p+A by A20,RLVECT_2:def 6;
   then Sum pL in Lin(-p+A) by RLVECT_3:14;
   then A21: Sum pL in Up Lin(-p+A);
   p+Sum pL=p+(1*(-p)+Sum L) by A19,Th39
    .=p+(-p+Sum L) by RLVECT_1:def 8
    .=(p+-p)+Sum L by RLVECT_1:def 3
    .=0.V+Sum L by RLVECT_1:5
    .=x by A18;
   hence thesis by A5,A21;
  end;
 end;
