
theorem Th36:
  for A,B being transitive with_units non empty AltCatStr,
  F being feasible reflexive FunctorStr over A,B st F is bijective coreflexive
  holds F" is reflexive
proof
  let A,B be transitive with_units non empty AltCatStr,
  F be feasible reflexive FunctorStr over A,B such that
A1: F is bijective and
A2: F is coreflexive;
  set G = F";
A3: the ObjectMap of G = (the ObjectMap of F)" by A1,Def38;
  let o be Object of B;
A4: dom the ObjectMap of F = [:the carrier of A,the carrier of A:]
  by FUNCT_2:def 1;
  consider p being Object of A such that
A5: o = F.p by A2,Th20;
  F is injective by A1;
  then F is one-to-one;
  then
A6: the ObjectMap of F is one-to-one;
A7: [p,p] in dom the ObjectMap of F by A4,ZFMISC_1:87;
A8: G.(F.p) = (G*F).p by Th33
    .= (((the ObjectMap of G)*the ObjectMap of F).(p,p))`1 by Def36
    .= ((id dom the ObjectMap of F).[p,p])`1 by A3,A6,FUNCT_1:39
    .= [p,p]`1 by A7,FUNCT_1:18
    .= p;
  thus (the ObjectMap of G).(o,o)
  = (the ObjectMap of G).((the ObjectMap of F).(p,p)) by A5,Def10
    .= ((the ObjectMap of G)*(the ObjectMap of F)).[p,p] by A7,FUNCT_1:13
    .= (id dom the ObjectMap of F).[p,p] by A3,A6,FUNCT_1:39
    .= [G.o,G.o] by A5,A7,A8,FUNCT_1:18;
end;
