reserve A,B,C,D,E for non empty set,
  a for Element of A,
  b for Element of B,
  c for Element of C,
  d for Element of D,
  X,Y,Z,S for set,x,y,z,s,t for object;

theorem Th5:
  for f1,f2 being Function of [:A,B,C,D:],E st for a,b,c,d holds f1
  .[a,b,c,d] = f2.[a,b,c,d] holds f1 = f2
proof
  let f1,f2 be Function of [:A,B,C,D:],E;
  assume for a,b,c,d holds f1.[a,b,c,d] = f2.[a,b,c,d];
  then
  for x,y,z,s st x in A & y in B & z in C & s in D holds f1.[x,y,z,s] = f2
  .[x,y,z,s];
  hence thesis by Th4;
end;
