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;
