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 Th64:
  L is convex & L.x = 1 implies Carrier L = {x}
 proof
  assume that
   A1: L is convex and
   A2: L.x=1;
  x in dom L by A2,FUNCT_1:def 2;
  then reconsider v=x as Element of V by FUNCT_2:def 1;
  consider K be Linear_Combination of{v} such that
   A3: K.v=jj by RLVECT_4:37;
  set LK=L-K;
  A4: Carrier K c={v} by RLVECT_2:def 6;
  sum LK=sum L-sum K by Th36
   .=sum L-1 by A3,A4,Th32
   .=1-1 by A1,Th62
   .=0;
  then consider F be FinSequence of V such that
   F is one-to-one and
   A5: rng F=Carrier LK and
   A6: 0=Sum(LK*F) by Def3;
  len(LK*F)=len F by FINSEQ_2:33;
  then A7: dom(LK*F)=dom F by FINSEQ_3:29;
  A8: for i be Nat st i in dom(LK*F) holds 0<=(LK*F).i
  proof
   let i be Nat;
   assume A9: i in dom(LK*F);
   then A10: (LK*F).i=LK.(F.i) by FUNCT_1:12;
   A11: F.i in Carrier LK by A5,A7,A9,FUNCT_1:def 3;
   then A12: LK.(F.i)=L.(F.i)-K.(F.i) by RLVECT_2:54;
   per cases;
   suppose F.i=v;
    hence thesis by A2,A3,A9,A12,FUNCT_1:12;
   end;
   suppose F.i<>v;
    then not F.i in Carrier K by A4,TARSKI:def 1;
    then K.(F.i)=0 by A11;
    hence thesis by A1,A10,A11,A12,Th62;
   end;
  end;
  Carrier LK={}
  proof
   assume Carrier LK<>{};
   then consider p be object such that
    A13: p in Carrier LK by XBOOLE_0:def 1;
   reconsider p as Element of V by A13;
   consider i be object such that
    A14: i in dom F and
    A15: F.i=p by A5,A13,FUNCT_1:def 3;
   reconsider i as Nat by A14;
   (LK*F).i>0
   proof
    A16: LK.p=L.p-K.p by RLVECT_2:54;
    per cases;
    suppose p=v;
     then LK.p=1-1 by A2,A3,RLVECT_2:54;
     hence thesis by A13,RLVECT_2:19;
    end;
    suppose p<>v;
     then not p in Carrier K by A4,TARSKI:def 1;
     then K.p=0;
     then A17: LK.p>=0 by A1,A16,Th62;
     LK.p<>0 by A13,RLVECT_2:19;
     hence thesis by A7,A14,A15,A17,FUNCT_1:12;
    end;
   end;
   hence thesis by A6,A7,A8,A14,RVSUM_1:85;
  end;
  then ZeroLC(V)=L+(-K) by RLVECT_2:def 5;
  then A18: -K=-L by RLVECT_2:50;
  A19: v in Carrier L by A2;
  -(-L)=L by RLVECT_2:53;
  then K=L by A18,RLVECT_2:53;
  hence thesis by A4,A19,ZFMISC_1:33;
 end;
