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
  x1 in A & x2 in A & 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
proof
  assume
A1: x1 in A & x2 in A & x1<>x2;
  consider f,g such that
A2: ( 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;
  let a,b;
  assume (ComplexFuncAdd(A)). ((ComplexFuncExtMult(A)).[a,f],(
  ComplexFuncExtMult(A)).[b,g]) = ComplexFuncZero(A);
  hence thesis by A1,A2,Th18;
end;
