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;

theorem Th22:
  A = {x1,x2} & x1<>x2 implies ex f,g st (for a,b st (
ComplexFuncAdd(A)).((ComplexFuncExtMult(A)).[a,f], (ComplexFuncExtMult(A)).[b,g
  ]) = ComplexFuncZero(A) holds a=0 & b=0) & for h holds ex a,b st h = (
ComplexFuncAdd(A)). ((ComplexFuncExtMult(A)).[a,f],(ComplexFuncExtMult(A)).[b,g
  ])
proof
  assume that
A1: A = {x1,x2} and
A2: x1<>x2;
  consider f,g such that
A3: ( for z st z in A holds (z=x1 implies f.z = 1r) & (z<>x1 implies f.z
=0c))& for z st z in A holds (z=x1 implies g.z = 0c) & (z<>x1 implies g.z = 1r)
  by Th17;
  take f,g;
  x1 in A & x2 in A by A1,TARSKI:def 2;
  hence thesis by A1,A2,A3,Th18,Th20;
end;
