reserve x1,x2,z for set;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);
reserve a,b for Real;

theorem
  ex V being strict RealLinearSpace st ex u,v being Element of V st (for
a,b st a*u + b*v = 0.V holds a=0 & b=0) & for w being Element of V ex a,b st w
  = a*u + b*v
proof
  set A ={0,1};
  take V = RealVectSpace(A);
  consider f,g being Element of Funcs(A,REAL) such that
A1: for a,b st (RealFuncAdd A).((RealFuncExtMult A).[a,f], (
  RealFuncExtMult A).[b,g]) = RealFuncZero(A) holds a=0 & b=0 and
A2: for h being Element of Funcs(A,REAL) holds ex a,b st h = (
RealFuncAdd A). ((RealFuncExtMult A).[a,f],(RealFuncExtMult A).[b,g]) by Th22;
  reconsider u=f, v=g as Element of V;
  take u,v;
  thus for a,b st a*u + b*v = 0.V holds a=0 & b=0 by A1;
  thus for w being Element of V ex a,b st w = a*u + b*v
  proof
    let w be Element of V;
    reconsider h=w as Element of Funcs(A,REAL);
    consider a,b such that
A3: h = (RealFuncAdd A).((RealFuncExtMult A).[a,f], (RealFuncExtMult A
    ).[b,g]) by A2;
    h = a*u + b*v by A3;
    hence thesis;
  end;
end;
