
theorem
  for V being RealLinearSpace, v1,v2,v3 being VECTOR of V st v1 <> v2 &
v1 <> v3 & v2 <> v3 holds ex L being Convex_Combination of V st for A being non
  empty Subset of V st {v1,v2,v3} c= A holds L is Convex_Combination of A
proof
  let V be RealLinearSpace;
  let v1,v2,v3 be VECTOR of V;
  assume that
A1: v1 <> v2 and
A2: v1 <> v3 and
A3: v2 <> v3;
  consider L being Linear_Combination of {v1,v2,v3} such that
A4: L.v1 = jj/3 & L.v2 = jj/3 & L.v3 = jj/3 by A1,A2,A3,RLVECT_4:39;
  consider F being FinSequence of the carrier of V such that
A5: 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
A6: len f = len F & for n being Nat st n in dom f holds f.n = F(n) from
  FINSEQ_1:sch 2;
  for x being object st x in {v1,v2,v3} holds x in Carrier(L)
  proof
    let x be object;
    assume
A7: x in {v1,v2,v3};
    then reconsider x as VECTOR of V;
    x = v1 or x = v2 or x = v3 by A7,ENUMSET1:def 1;
    hence thesis by A4,RLVECT_2:19;
  end;
  then Carrier(L) c= {v1,v2,v3} & {v1,v2,v3} c= Carrier(L) by RLVECT_2:def 6;
  then
A8: {v1,v2,v3} = Carrier(L) by XBOOLE_0:def 10;
  then
A9: len F = 3 by A1,A2,A3,A5,FINSEQ_3:101;
  then 2 in dom f by A6,FINSEQ_3:25;
  then
A10: f.2 = L.(F.2) by A6;
  3 in dom f by A6,A9,FINSEQ_3:25;
  then
A11: f.3 = L.(F.3) by A6;
  1 in dom f by A6,A9,FINSEQ_3:25;
  then
A12: f.1 = L.(F.1) by A6;
  now
    per cases by A1,A2,A3,A8,A5,CONVEX1:31;
    suppose
A13:  F = <*v1,v2,v3*>;
      then
A14:  F.3 = v3 by FINSEQ_1:45;
A15:  F.1 = v1 & F.2 = v2 by A13,FINSEQ_1:45;
      then f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A14,FINSEQ_1:45;
      then f = <*jt*>^<*jt*>^<*jt*> by FINSEQ_1:def 10;
      then rng f = rng (<*jt*>^<*jt*>) \/ rng <*1/3*> by FINSEQ_1:31
        .= rng <*1/3*> \/ rng <*jt*> \/ rng <*jt*> by FINSEQ_1:31
        .= {jt} by FINSEQ_1:38;
      then reconsider f as FinSequence of REAL by FINSEQ_1:def 4;
A16:  for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
      proof
        let n be Nat;
        assume
A17:    n in dom f;
        then n in Seg len f by FINSEQ_1:def 3;
        hence thesis by A4,A6,A9,A12,A10,A11,A15,A14,A17,ENUMSET1:def 1
,FINSEQ_3:1;
      end;
      f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A15,A14,FINSEQ_1:45;
      then Sum(f) = 1/3 + 1/3 + 1/3 by RVSUM_1:78
        .= 1;
      then reconsider L as Convex_Combination of V by A5,A6,A16,CONVEX1:def 3;
      take L;
      for A being non empty Subset of V st {v1,v2,v3} c= A holds L is
      Convex_Combination of A by A8,RLVECT_2:def 6;
      hence thesis;
    end;
    suppose
A18:  F = <* v1,v3,v2*>;
      then
A19:  F.3 = v2 by FINSEQ_1:45;
A20:  F.1 = v1 & F.2 = v3 by A18,FINSEQ_1:45;
      then f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A19,FINSEQ_1:45;
      then f = <*jt*>^<*jt*>^<*jt*> by FINSEQ_1:def 10;
      then rng f = rng (<*jt*>^<*jt*>) \/ rng <*jt*> by FINSEQ_1:31
        .= rng <*jt*> \/ rng <*jt*> \/ rng <*jt*> by FINSEQ_1:31
        .= {jt} by FINSEQ_1:38;
      then reconsider f as FinSequence of REAL by FINSEQ_1:def 4;
A21:  for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
      proof
        let n be Nat;
        assume
A22:    n in dom f;
        then n in Seg len f by FINSEQ_1:def 3;
        hence thesis by A4,A6,A9,A12,A10,A11,A20,A19,A22,ENUMSET1:def 1
,FINSEQ_3:1;
      end;
      f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A20,A19,FINSEQ_1:45;
      then Sum(f) = 1/3 + 1/3 + 1/3 by RVSUM_1:78
        .= 1;
      then reconsider L as Convex_Combination of V by A5,A6,A21,CONVEX1:def 3;
      take L;
      for A being non empty Subset of V st {v1,v2,v3} c= A holds L is
      Convex_Combination of A by A8,RLVECT_2:def 6;
      hence thesis;
    end;
    suppose
A23:  F = <*v2,v1,v3*>;
      then
A24:  F.3 = v3 by FINSEQ_1:45;
A25:  F.1 = v2 & F.2 = v1 by A23,FINSEQ_1:45;
      then f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A24,FINSEQ_1:45;
      then f = <*jt*>^<*jt*>^<*jt*> by FINSEQ_1:def 10;
      then rng f = rng (<*jt*>^<*jt*>) \/ rng <*jt*> by FINSEQ_1:31
        .= rng <*jt*> \/ rng <*jt*> \/ rng <*jt*> by FINSEQ_1:31
        .= {jt} by FINSEQ_1:38;
      then reconsider f as FinSequence of REAL by FINSEQ_1:def 4;
A26:  for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
      proof
        let n be Nat;
        assume
A27:    n in dom f;
        then n in Seg len f by FINSEQ_1:def 3;
        hence thesis by A4,A6,A9,A12,A10,A11,A25,A24,A27,ENUMSET1:def 1
,FINSEQ_3:1;
      end;
      f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A25,A24,FINSEQ_1:45;
      then Sum(f) = 1/3 + 1/3 + 1/3 by RVSUM_1:78
        .= 1;
      then reconsider L as Convex_Combination of V by A5,A6,A26,CONVEX1:def 3;
      take L;
      for A being non empty Subset of V st {v1,v2,v3} c= A holds L is
      Convex_Combination of A by A8,RLVECT_2:def 6;
      hence thesis;
    end;
    suppose
A28:  F = <* v2,v3,v1*>;
      then
A29:  F.3 = v1 by FINSEQ_1:45;
A30:  F.1 = v2 & F.2 = v3 by A28,FINSEQ_1:45;
      then f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A29,FINSEQ_1:45;
      then f = <*jt*>^<*jt*>^<*jt*> by FINSEQ_1:def 10;
      then rng f = rng (<*jt*>^<*jt*>) \/ rng <*jt*> by FINSEQ_1:31
        .= rng <*jt*> \/ rng <*jt*> \/ rng <*jt*> by FINSEQ_1:31
        .= {jt} by FINSEQ_1:38;
      then reconsider f as FinSequence of REAL by FINSEQ_1:def 4;
A31:  for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
      proof
        let n be Nat;
        assume
A32:    n in dom f;
        then n in Seg len f by FINSEQ_1:def 3;
        hence thesis by A4,A6,A9,A12,A10,A11,A30,A29,A32,ENUMSET1:def 1
,FINSEQ_3:1;
      end;
      f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A30,A29,FINSEQ_1:45;
      then Sum(f) = 1/3 + 1/3 + 1/3 by RVSUM_1:78
        .= 1;
      then reconsider L as Convex_Combination of V by A5,A6,A31,CONVEX1:def 3;
      take L;
      for A being non empty Subset of V st {v1,v2,v3} c= A holds L is
      Convex_Combination of A by A8,RLVECT_2:def 6;
      hence thesis;
    end;
    suppose
A33:  F = <* v3,v1,v2*>;
      then
A34:  F.3 = v2 by FINSEQ_1:45;
A35:  F.1 = v3 & F.2 = v1 by A33,FINSEQ_1:45;
      then f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A34,FINSEQ_1:45;
      then f = <*jt*>^<*jt*>^<*jt*> by FINSEQ_1:def 10;
      then rng f = rng (<*jt*>^<*jt*>) \/ rng <*jt*> by FINSEQ_1:31
        .= rng <*jt*> \/ rng <*jt*> \/ rng <*jt*> by FINSEQ_1:31
        .= {jt} by FINSEQ_1:38;
      then reconsider f as FinSequence of REAL by FINSEQ_1:def 4;
A36:  for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
      proof
        let n be Nat;
        assume
A37:    n in dom f;
        then n in Seg len f by FINSEQ_1:def 3;
        hence thesis by A4,A6,A9,A12,A10,A11,A35,A34,A37,ENUMSET1:def 1
,FINSEQ_3:1;
      end;
      f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A35,A34,FINSEQ_1:45;
      then Sum(f) = 1/3 + 1/3 + 1/3 by RVSUM_1:78
        .= 1;
      then reconsider L as Convex_Combination of V by A5,A6,A36,CONVEX1:def 3;
      take L;
      for A being non empty Subset of V st {v1,v2,v3} c= A holds L is
      Convex_Combination of A by A8,RLVECT_2:def 6;
      hence thesis;
    end;
    suppose
A38:  F = <* v3,v2,v1*>;
      then
A39:  F.3 = v1 by FINSEQ_1:45;
A40:  F.1 = v3 & F.2 = v2 by A38,FINSEQ_1:45;
      then f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A39,FINSEQ_1:45;
      then f = <*jt*>^<*jt*>^<*jt*> by FINSEQ_1:def 10;
      then rng f = rng (<*jt*>^<*jt*>) \/ rng <*jt*> by FINSEQ_1:31
        .= rng <*jt*> \/ rng <*jt*> \/ rng <*jt*> by FINSEQ_1:31
        .= {jt} by FINSEQ_1:38;
      then reconsider f as FinSequence of REAL by FINSEQ_1:def 4;
A41:  for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
      proof
        let n be Nat;
        assume
A42:    n in dom f;
        then n in Seg len f by FINSEQ_1:def 3;
        hence thesis by A4,A6,A9,A12,A10,A11,A40,A39,A42,ENUMSET1:def 1
,FINSEQ_3:1;
      end;
      f = <*1/3,1/3,1/3*> by A4,A6,A9,A12,A10,A11,A40,A39,FINSEQ_1:45;
      then Sum(f) = 1/3 + 1/3 + 1/3 by RVSUM_1:78
        .= 1;
      then reconsider L as Convex_Combination of V by A5,A6,A41,CONVEX1:def 3;
      take L;
      for A being non empty Subset of V st {v1,v2,v3} c= A holds L is
      Convex_Combination of A by A8,RLVECT_2:def 6;
      hence thesis;
    end;
  end;
  hence thesis;
end;
