reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;
reserve v,v1,v2,u,w for Vector of n-VectSp_over K,
  t,t1,t2 for Element of n -tuples_on the carrier of K,

  L for Linear_Combination of n-VectSp_over K,
  M,M1 for Matrix of m,n,K;

theorem Th116:
  for V be VectSp of K for u, v be Vector of V holds x in Lin {u,
  v} iff ex a,b st x = a * u + b * v
proof
  let V be VectSp of K;
  let u, v be Vector of V;
  per cases;
  suppose
A1: u=v;
    then
A2: {u,v}={u} by ENUMSET1:29;
    thus x in Lin {u,v} implies ex a,b st x = a * u + b * v
    proof
      assume x in Lin{u,v};
      then consider a such that
A3:   x=a*u by A2,VECTSP10:3;
      x = a*u+0.V by A3,RLVECT_1:def 4
        .= a*u+0.K*v by VECTSP10:1;
      hence thesis;
    end;
    given a,b such that
A4: x = a * u + b * v;
    x=(a+b)*u by A1,A4,VECTSP_1:def 15;
    hence thesis by A2,VECTSP10:3;
  end;
  suppose
A5: u<>v;
    thus x in Lin {u,v} implies ex a,b st x = a * u + b * v
    proof
      assume x in Lin {u,v};
      then consider L be Linear_Combination of {u,v} such that
A6:   x=Sum L by VECTSP_7:7;
      x=L.u*u+L.v*v by A5,A6,VECTSP_6:18;
      hence thesis;
    end;
    deffunc F(set) = 0.K;
    given a,b such that
A7: x = a * u + b * v;
    consider L be Function of the carrier of V, the carrier of K such that
A8: L.u = a & L.v=b and
A9: for z be Element of V st z <> u & z<>v holds L.z = F(z) from
    FUNCT_2:sch 7(A5);
    reconsider L as Element of Funcs(the carrier of V,the carrier of K) by
FUNCT_2:8;
    now
      let z be Vector of V such that
A10:  not z in {u,v};
A11:  z<>u by A10,TARSKI:def 2;
      z <> v by A10,TARSKI:def 2;
      hence L.z = 0.K by A9,A11;
    end;
    then reconsider L as Linear_Combination of V by VECTSP_6:def 1;
    Carrier L c= {u,v}
    proof
      let x be object such that
A12:  x in Carrier L;
      L.x <> 0.K by A12,VECTSP_6:2;
      then x = v or x=u by A9,A12;
      hence thesis by TARSKI:def 2;
    end;
    then reconsider L as Linear_Combination of {u,v} by VECTSP_6:def 4;
    Sum(L) = x by A5,A7,A8,VECTSP_6:18;
    hence thesis by VECTSP_7:7;
  end;
end;
