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 Th16:
  ex V being non trivial RealLinearSpace st ex u,v,w being Element
  of V st (for a,b,c st a*u + b*v + c*w = 0.V holds a=0 & b=0 & c = 0) & for y
  being Element of V ex a,b,c st y = a*u + b*v + c*w
proof
  consider A,x1,x2,x3 such that
A1: A={x1,x2,x3} & x1<>x2 & x1<>x3 & x2<>x3 by Lm30;
  set V = RealVectSpace(A);
  consider f,g,h such that
A2: for a,b,c being Real
   st (RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A
  )). [a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,h]) =
  RealFuncZero(A ) holds a=0 & b=0 & c = 0 and
A3: for h9 being Element of Funcs(A,REAL)
  ex a,b,c being Real st h9 = (
RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult
  (A)).[b,g]), (RealFuncExtMult(A)).[c,h]) by A1,Th15;
  reconsider u=f, v=g, w = h as Element of V;
  for a,b,c st a*u + b*v + c*w = 0.V holds a=0 & b=0 & c = 0
   by A2;
  then u is not zero by Th1;
  then
A4: u <> 0.V;
A5: for y being Element of V ex a,b,c st y = a*u + b*v + c*w
  proof
    let y be Element of V;
    reconsider h9=y as Element of Funcs(A,REAL);
    consider a,b,c being Real such that
A6: h9 = (RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,
    f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,h]) by A3;
    h9 = a*u + b*v + c*w by A6;
    hence thesis;
  end;
  reconsider V as non trivial RealLinearSpace by A4,STRUCT_0:def 18;
  take V;
  reconsider u,v,w as Element of V;
  take u,v,w;
  thus for a,b,c st a*u + b*v + c*w = 0.V holds a=0 & b=0 & c = 0
   by A2;
  let y be Element of V;
   ex a,b,c st y = a*u + b*v + c*w by A5;
  hence thesis;
end;
