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);
reserve l,l1,l2,l3 for Element of MapsC(X);
reserve T,T1,T2 for Element of TOL(X);
reserve f for Element of FuncsT(X);

theorem Th38:
  x in FuncsT(X) iff ex T1,T2 st (T2`2 = {} implies T1`2 = {}) & x
  is Function of T1`2,T2`2
proof
  set F = the set of all
 Funcs(T`2,TT`2) where T is Element of TOL(X), TT is Element of TOL
  (X);
  thus x in FuncsT(X) implies ex A,B be Element of TOL(X) st (B`2={} implies A
  `2={}) & x is Function of A`2,B`2
  proof
    assume x in FuncsT(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 TOL(X) such that
A3: C = Funcs(A`2,B`2) by A2;
    take A,B;
    thus thesis by A1,A3,FUNCT_2:66;
  end;
  given A,B be Element of TOL(X) such that
A4: ( B`2={} implies A`2={})& x is Function of A`2,B`2;
A5: Funcs(A`2,B`2) in F;
  x in Funcs(A`2,B`2) by A4,FUNCT_2:8;
  hence thesis by A5,TARSKI:def 4;
end;
