
theorem Th24:
  for C1,C2 being non empty AltGraph, o2 being Object of C2 st <^o2,o2^> <> {}
  for m be Morphism of o2,o2, o,o9 being Object of C1, f being Morphism of o,o9
  st <^o,o9^> <> {} holds Morph-Map(C1 --> m,o,o9).f = m
proof
  let C1,C2 be non empty AltGraph, o2 be Object of C2 such that
A1: <^o2,o2^> <> {};
  let m be Morphism of o2,o2, o,o9 be Object of C1, f be Morphism of o,o9
  such that
A2: <^o,o9^> <> {};
  set X =
  the set of all
 [[o1,o19],<^o1,o19^> --> m] where o1 is Object of C1, o19 is Object of C1;
 set Y = the set of all  [[o1,o19],(the Arrows of C1).(o1,o19) --> m]
  where o1 is Element of C1, o19 is Element of C1;
A3: X c= Y
  proof
    let e be object;
    assume e in X;
    then consider o1,o19 being Object of C1 such that
A4: e = [[o1,o19],<^o1,o19^> --> m];
    e = [[o1,o19],(the Arrows of C1).(o1,o19) --> m] by A4,ALTCAT_1:def 1;
    hence thesis;
  end;
A5: Y c= X
  proof
    let e be object;
    assume e in Y;
    then consider o1,o19 being Element of C1 such that
A6: e = [[o1,o19],(the Arrows of C1).(o1,o19) --> m];
    reconsider o1,o19 as Object of C1;
    e = [[o1,o19],<^o1,o19^> --> m] by A6,ALTCAT_1:def 1;
    hence thesis;
  end;
  defpred P[set,set] means not contradiction;
  deffunc F(Element of C1,Element of C1) = (the Arrows of C1).($1,$2) --> m;
  the MorphMap of C1 --> m = X by A1,Def17;
  then
A7: the MorphMap of C1 --> m = { [[o1,o19],F(o1,o19)]
  where o1 is Element of C1, o19 is Element of C1: P[o1,o19] }
  by A3,A5;
A8: P[o,o9];
  Morph-Map(C1 --> m,o,o9) = (the MorphMap of C1 --> m).(o,o9)
    .= F(o,o9) from ValOnPair(A7,A8);
  hence Morph-Map(C1 --> m,o,o9).f = (<^o,o9^> --> m).f by ALTCAT_1:def 1
    .= m by A2,FUNCOP_1:7;
end;
