reserve
  I for set,
  E for non empty set;

theorem Th4:
  for A being ObjectsFamily of I,EnsCat {{}},
      o being Object of EnsCat {{}},
      P being MorphismsFamily of o,A st P = I --> {} holds
   P is feasible projection-morphisms
   proof
     let A be ObjectsFamily of I,EnsCat {{}};
     let o be Object of EnsCat {{}};
     let P be MorphismsFamily of o,A;
     assume
A1:  P = I --> {};
     thus P is feasible
     proof
       let i be set;
       assume
A2:    i in I;
       then reconsider I as non empty set;
       reconsider i as Element of I by A2;
       reconsider A as ObjectsFamily of I,C;
       P.i = {} by A1;
       then P.i in <^o,A.i^> by Lm3;
       hence thesis;
     end;
     let Y be Object of C, F being MorphismsFamily of Y,A;
     assume F is feasible;
     reconsider f = {} as Morphism of Y,o by Lm3;
     take f;
     thus f in <^Y,o^> by Lm3;
     thus for i being set st i in I
     ex si being Object of C, Pi being Morphism of o,si st
     si = A.i & Pi = P.i & F.i = Pi * f
     proof
       let i be set;
       assume
A3:    i in I;
       then reconsider I as non empty set;
       reconsider j = i as Element of I by A3;
       reconsider M = {} as Morphism of o,o by Lm3;
       reconsider A1 = A as ObjectsFamily of I,C;
       reconsider F1 = F as MorphismsFamily of Y,A1;
       take o, M;
       A1.j = {} by Lm1,TARSKI:def 1;
       hence o = A.i by Lm5;
       thus M = P.i by A1;
       F1.j is Morphism of Y,o & M*f is Morphism of Y,o by Lm5;
       hence thesis by Lm6;
     end;
     thus thesis by Lm4;
   end;
