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 Th40:
  for f being Function of T1`2,T2`2 st (T2`2={} implies T1`2={}) &
(for x,y st [x,y] in T1`1 holds [f.x,f.y] in T2`1) holds [[T1,T2],f] in MapsT(X
  )
proof
  let f be Function of T1`2,T2`2;
  assume that
A1: T2`2={} implies T1`2={} and
A2: for x,y st [x,y] in T1`1 holds [f.x,f.y] in T2`1;
  reconsider f9 = f as Element of FuncsT(X) by A1,Th38;
  for x,y st [x,y] in T1`1 holds [f9.x,f9.y] in T2`1 by A2;
  hence thesis by A1;
end;
