reserve x1,x2,z for set;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,COMPLEX);
reserve a,b for Complex;
reserve C for strict non empty CLSStruct,
        u,v,w for Element of C;

theorem
  ex V being strict ComplexLinearSpace st ex u,v being VECTOR of V st (
  for a,b st a*u + b*v = 0.V holds a=0 & b=0) & for w being VECTOR of V
  ex a,b st w = a*u + b*v
proof
  consider A,x1,x2 such that
A1: A={x1,x2} & x1<>x2 by Lm3;
  take V = ComplexVectSpace(A);
  consider f,g such that
A2: for a,b st (ComplexFuncAdd(A)).((ComplexFuncExtMult(A)).[a,f], (
  ComplexFuncExtMult(A)).[b,g]) = ComplexFuncZero(A) holds a=0c & b=0c and
A3: for h holds ex a,b st h = (ComplexFuncAdd(A)). ((ComplexFuncExtMult(
  A)).[a,f],(ComplexFuncExtMult(A)).[b,g]) by A1,Th22;
  reconsider u=f, v=g as VECTOR of V;
  take u,v;
  thus for a,b st a*u + b*v = 0.V holds a=0 & b=0 by A2;
  thus for w being VECTOR of V ex a,b st w = a*u + b*v
  proof
    let w be VECTOR of V;
    reconsider h=w as Element of Funcs(A,COMPLEX);
    consider a,b such that
A4: h = (ComplexFuncAdd(A)).((ComplexFuncExtMult(A)).[a,f], (
    ComplexFuncExtMult(A)).[b,g]) by A3;
    h = a*u + b*v by A4;
    hence thesis;
  end;
end;
