
theorem
  for V being RealLinearSpace, v1,v2 being VECTOR of V st v1 <> v2 holds
ex L being Convex_Combination of V st for A being non empty Subset of V st {v1,
  v2} c= A holds L is Convex_Combination of A
proof
  let V be RealLinearSpace;
  let v1,v2 be VECTOR of V;
  assume
A1: v1 <> v2;
  consider L being Linear_Combination of {v1,v2} such that
A2: L.v1 = jj/2 & L.v2 = jj/2 by A1,RLVECT_4:38;
  consider F being FinSequence of the carrier of V such that
A3: F is one-to-one & rng F = Carrier(L) and
  Sum(L) = Sum(L (#) F) by RLVECT_2:def 8;
  deffunc F(set) = L.(F.$1);
  consider f being FinSequence such that
A4: len f = len F & for n being Nat st n in dom f holds f.n = F(n) from
  FINSEQ_1:sch 2;
  v1 in Carrier(L) & v2 in Carrier(L) by A2,RLVECT_2:19;
  then Carrier(L) c= {v1,v2} & {v1,v2} c= Carrier(L) by RLVECT_2:def 6
,ZFMISC_1:32;
  then
A5: {v1,v2} = Carrier(L) by XBOOLE_0:def 10;
  then
A6: len F = 2 by A1,A3,FINSEQ_3:98;
  then 2 in dom f by A4,FINSEQ_3:25;
  then
A7: f.2 = L.(F.2) by A4;
  1 in dom f by A4,A6,FINSEQ_3:25;
  then
A8: f.1 = L.(F.1) by A4;
  now
    per cases by A1,A5,A3,FINSEQ_3:99;
    suppose
      F = <*v1,v2*>;
      then
A9:   F.1 = v1 & F.2 = v2 by FINSEQ_1:44;
      then f = <*1/2,1/2*> by A2,A4,A6,A8,A7,FINSEQ_1:44;
      then f = <*jd*>^<*jd*> by FINSEQ_1:def 9;
      then rng f = rng <*1/2*> \/ rng <*jd*> by FINSEQ_1:31
        .= {jd} by FINSEQ_1:38;
      then reconsider f as FinSequence of REAL by FINSEQ_1:def 4;
A10:  for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
      proof
        let n be Nat;
        assume
A11:    n in dom f;
        then n in Seg len f by FINSEQ_1:def 3;
        hence thesis by A2,A4,A6,A8,A7,A9,A11,FINSEQ_1:2,TARSKI:def 2;
      end;
      f = <*1/2,1/2*> by A2,A4,A6,A8,A7,A9,FINSEQ_1:44;
      then Sum(f) = 1/2 + 1/2 by RVSUM_1:77
        .= 1;
      then reconsider L as Convex_Combination of V by A3,A4,A10,CONVEX1:def 3;
      take L;
      for A being non empty Subset of V st {v1,v2} c= A holds L is
      Convex_Combination of A by A5,RLVECT_2:def 6;
      hence thesis;
    end;
    suppose
      F = <*v2,v1*>;
      then
A12:  F.1 = v2 & F.2 = v1 by FINSEQ_1:44;
      then f = <*1/2,1/2*> by A2,A4,A6,A8,A7,FINSEQ_1:44;
      then f = <*jd*>^<*jd*> by FINSEQ_1:def 9;
      then rng f = rng <*1/2*> \/ rng <*jd*> by FINSEQ_1:31
        .= {jd} by FINSEQ_1:38;
      then reconsider f as FinSequence of REAL by FINSEQ_1:def 4;
A13:  for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
      proof
        let n be Nat;
        assume
A14:    n in dom f;
        then n in Seg len f by FINSEQ_1:def 3;
        hence thesis by A2,A4,A6,A8,A7,A12,A14,FINSEQ_1:2,TARSKI:def 2;
      end;
      f = <*1/2,1/2*> by A2,A4,A6,A8,A7,A12,FINSEQ_1:44;
      then Sum(f) = 1/2 + 1/2 by RVSUM_1:77
        .= 1;
      then reconsider L as Convex_Combination of V by A3,A4,A13,CONVEX1:def 3;
      take L;
      for A being non empty Subset of V st {v1,v2} c= A holds L is
      Convex_Combination of A by A5,RLVECT_2:def 6;
      hence thesis;
    end;
  end;
  hence thesis;
end;
