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
  x1<>x2 & x1<>x3 & x2<>x3 implies ex f,g,h st
 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
proof
  assume
A1: x1<>x2 & x1<>x3 & x2<>x3;
  consider f such that
A2: f.x1 = 1 & for z st z in A holds (z<>x1 implies f.z = 0) by Th10;
  consider h such that
A3: h.x3 = 1 & for z st z in A holds(z<>x3 implies h.z = 0) by Th10;
  consider g such that
A4: g.x2 = 1 & for z st z in A holds(z<>x2 implies g.z = 0) by Th10;
  take f,g,h;
  let a,b,c be Real;
  assume (RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(
  RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,h]) = RealFuncZero(A);
  hence thesis by A1,A2,A4,A3,Th11;
end;
