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);
reserve m,m1,m2,m3 for Element of MapsT(X);

theorem Th39:
  ex f,T1,T2 st m = [[T1,T2],f] & (T2`2 = {} implies T1`2 = {}) &
  f is Function of T1`2,T2`2 & for x,y st [x,y] in T1`1 holds [f.x,f.y] in T2`1
proof
  m in {[[T1,T2],f]: (T2`2={} implies T1`2={}) & f is Function of T1`2,T2
  `2 & for x,y st [x,y] in T1`1 holds [f.x,f.y] in T2`1};
  then ex T1,T2,f st m = [[T1,T2],f] & (T2`2={} implies T1`2={}) & f is
  Function of T1`2,T2`2 & for x,y st [x,y] in T1`1 holds [f.x,f.y] in T2`1;
  hence thesis;
end;
