reserve C for category,
  o1, o2, o3 for Object of C;

theorem Th15:
  for C1, C2 being non empty AltCatStr for F being Contravariant
  FunctorStr over C1, C2 holds F is faithful iff for o1, o2 being Object of C1
  holds Morph-Map(F,o2,o1) is one-to-one
proof
  let C1, C2 be non empty AltCatStr, F be Contravariant FunctorStr over C1,C2;
  set I = [:the carrier of C1, the carrier of C1:];
  hereby
    assume F is faithful;
    then
A1: (the MorphMap of F) is "1-1";
    let o1, o2 be Object of C1;
    [o2,o1] in I & dom(the MorphMap of F) = I by PARTFUN1:def 2,ZFMISC_1:87;
    hence Morph-Map(F,o2,o1) is one-to-one by A1;
  end;
  assume
A2: for o1, o2 being Object of C1 holds Morph-Map(F,o2,o1) is one-to-one;
  let i be set, f be Function such that
A3: i in dom(the MorphMap of F) and
A4: (the MorphMap of F).i = f;
  dom(the MorphMap of F) = I by PARTFUN1:def 2;
  then consider o1, o2 being object such that
A5: o1 in the carrier of C1 & o2 in the carrier of C1 and
A6: i = [o1,o2] by A3,ZFMISC_1:84;
  reconsider o1, o2 as Object of C1 by A5;
  (the MorphMap of F).(o1,o2) = Morph-Map(F,o1,o2);
  hence thesis by A2,A4,A6;
end;
