reserve x1,x2,z for set;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);
reserve a,b for Real;

theorem Th22:
  A = {x1,x2} & x1<>x2 implies ex f,g st (for a,b st (RealFuncAdd
  A).((RealFuncExtMult A).[a,f], (RealFuncExtMult A).[b,g]) = RealFuncZero(A)
holds a=0 & b=0) & for h holds ex a,b st h = (RealFuncAdd A). ((RealFuncExtMult
  A).[a,f],(RealFuncExtMult A).[b,g])
proof
  assume that
A1: A = {x1,x2} and
A2: x1<>x2;
  x1 in A by TARSKI:def 2,A1; then
  reconsider f = (RealFuncZero A) +* (x1 .--> 1),
    g = (RealFuncUnit A) +* (x1 .--> 0) as Element of Funcs (A,REAL) by Th17;
  take f,g;
  x1 in A & x2 in A by A1,TARSKI:def 2;
  hence thesis by A1,A2,Th18,Th20;
end;
