reserve V for RealLinearSpace,
  o,p,q,r,s,u,v,w,y,y1,u1,v1,w1,u2,v2,w2 for Element of V,
  a,b,c,d,a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3 for Real,
  z for set;
reserve A for non empty set;
reserve f,g,h,f1 for Element of Funcs(A,REAL);
reserve x1,x2,x3,x4 for Element of A;

theorem Th22:
  ex V being non trivial RealLinearSpace st ex u,v,w,u1 being
Element of V st (for a,b,c,d st a*u + b*v + c*w + d*u1 = 0.V holds a = 0 & b =
0 & c = 0 & d = 0) & for y being Element of V ex a,b,c,d st y = a*u + b*v + c*w
  + d*u1
proof
  consider A,x1,x2,x3,x4 such that
A1: A={x1,x2,x3,x4} & x1<>x2 & x1<>x3 & x1<>x4 & x2<>x3 & x2<>x4 & x3 <>
  x4 by Lm31;
  set V = RealVectSpace(A);
  consider f,g,h,f1 such that
A2: for a,b,c,d being Real
  st (RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)).
(( RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).
[c, h]),(RealFuncExtMult(A)).[d,f1]) = RealFuncZero(A) holds a = 0 & b = 0 & c
  = 0 & d = 0 and
A3: for h9 being Element of Funcs(A,REAL)
  ex a,b,c,d being Real st h9 = (
RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f
],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,h]),(RealFuncExtMult(A))
  .[d,f1]) by A1,Th21;
  reconsider u = f, v = g, w = h, u1 = f1 as Element of V;
  for a,b,c,d st a*u + b*v + c*w + d*u1 = 0.V holds a = 0 & b = 0 & c = 0
  & d = 0
  by A2;
  then u is not zero by Th2;
  then
A4: u <> 0.V;
A5: for y being Element of V ex a,b,c,d st y = a*u + b*v + c*w + d*u1
  proof
    let y be Element of V;
    reconsider h9=y as Element of Funcs(A,REAL);
    consider a,b,c,d being Real such that
A6: h9 = (RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)). ((
RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,
    h]),(RealFuncExtMult(A)).[d,f1]) by A3;
    h9 = a * u + b*v + c*w + d*u1 by A6;
    hence thesis;
  end;
  reconsider V as non trivial RealLinearSpace by A4,STRUCT_0:def 18;
  take V;
  reconsider u,v,w,u1 as Element of V;
  take u,v,w,u1;
  thus for a,b,c,d st a*u + b*v + c*w + d*u1 = 0.V holds a = 0 & b =
0 & c = 0 & d = 0
   by A2;
  let y be Element of V;
   ex a,b,c,d st y = a*u + b*v + c*w + d*u1 by A5;
  hence thesis;
end;
