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 Th10:
  ex f st f.x1 = 1 & for z st z in A & z<>x1 holds f.z = 0
proof
  deffunc G(object) = In(0,REAL);
  deffunc F(object) = 1;
  defpred P[object] means $1 = x1;
A1: for z being object st z in A
holds (P[z] implies F(z) in REAL) & (not P[z] implies G(
  z) in REAL) by XREAL_0:def 1;
  consider f being Function of A,REAL such that
A2: for z being object st z in A
holds (P[z] implies f.z = F(z)) & (not P[z] implies
  f.z = G(z)) from FUNCT_2:sch 5(A1);
  reconsider f as Element of Funcs(A,REAL) by FUNCT_2:8;
  take f;
  thus thesis by A2;
end;
