reserve x,y,z,a,b,c,X,A for set;
reserve C,D for Coherence_Space;
reserve T for Tolerance of union C;
reserve E for Tolerance of X;
reserve C,C1,C2 for Element of CSp(X);
reserve g for Element of FuncsC(X);

theorem Th17:
  x in FuncsC(X) iff ex C1,C2 st (union C2 = {} implies union C1 =
  {}) & x is Function of union C1,union C2
proof
  set F = the set of all
 Funcs(union xx,union y) where xx is Element of CSp(X), y is
  Element of CSp(X);
  thus x in FuncsC(X) implies ex C1,C2 st (union C2 = {} implies union C1 = {}
  ) & x is Function of union C1,union C2
  proof
    assume x in FuncsC(X);
    then consider C being set such that
A1: x in C and
A2: C in F by TARSKI:def 4;
    consider A,B be Element of CSp(X) such that
A3: C = Funcs(union A,union B) by A2;
    take A,B;
    thus thesis by A1,A3,FUNCT_2:66;
  end;
  given A,B be Element of CSp(X) such that
A4: ( union B={} implies union A={})& x is Function of union A,union B;
A5: Funcs(union A,union B) in F;
  x in Funcs(union A,union B) by A4,FUNCT_2:8;
  hence thesis by A5,TARSKI:def 4;
end;
