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 Th17:
  ex f,g st (for z st z in A holds (z = x1 implies f.z = 1r) & (z
<>x1 implies f.z = 0)) & for z st z in A holds (z = x1 implies g.z = 0) & (z<>
  x1 implies g.z = 1r)
proof
  deffunc G(object) = 1r;
  deffunc F(object) = 0c;
  defpred P[object] means $1 = x1;
A1: for z being object st z in A
holds (P[z] implies G(z) in COMPLEX) & (not P[z] implies
  F(z) in COMPLEX);
  consider f being Function of A,COMPLEX such that
A2: for z being object st z in A
 holds (P[z] implies f.z = G(z)) & (not P[z] implies
  f.z = F(z)) from FUNCT_2:sch 5(A1);
A3: for z being object st z in A
holds (P[z] implies F(z) in COMPLEX) & (not P[z] implies
  G(z) in COMPLEX);
  consider g being Function of A,COMPLEX such that
A4: for z being object st z in A
holds (P[z] implies g.z = F(z)) & (not P[z] implies
  g.z = G(z)) from FUNCT_2:sch 5(A3);
  reconsider f,g as Element of Funcs(A,COMPLEX) by FUNCT_2:8;
  take f,g;
  thus thesis by A2,A4;
end;
