
theorem Th23:
  for C1, C2 being non empty AltGraph,
  F being Contravariant FunctorStr over C1,C2, o1,o2 being Object of C1
  holds (the ObjectMap of F).(o1,o2) = [F.o2,F.o1]
proof
  let C1, C2 be non empty AltGraph, F be Contravariant FunctorStr over C1,C2,
  o1,o2 be Object of C1;
  the ObjectMap of F is Contravariant by Def13;
  then consider f being Function of the carrier of C1, the carrier of C2 such
  that
A1: the ObjectMap of F = ~[:f,f:];
A2: dom[:f,f:] = [:the carrier of C1, the carrier of C1:] by FUNCT_2:def 1;
  then [o1,o1] in dom[:f,f:] by ZFMISC_1:87;
  then
A3: F.o1 = ([:f,f:].(o1,o1))`1 by A1,FUNCT_4:def 2
    .= ([f.o1,f.o1])`1 by FUNCT_3:75
    .= f.o1;
  [o2,o2] in dom[:f,f:] by A2,ZFMISC_1:87;
  then
A4: F.o2 = ([:f,f:].(o2,o2))`1 by A1,FUNCT_4:def 2
    .= ([f.o2,f.o2])`1 by FUNCT_3:75
    .= f.o2;
  [o2,o1] in dom[:f,f:] by A2,ZFMISC_1:87;
  hence (the ObjectMap of F).(o1,o2) = [:f,f:].(o2,o1) by A1,FUNCT_4:def 2
    .= [F.o2,F.o1] by A3,A4,FUNCT_3:75;
end;
