reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;
reserve F,F1,F2,F3 for Functor of A,B,
  G,G1,G2,G3 for Functor of B,C,
  H,H1,H2 for Functor of C,D,
  s for natural_transformation of F1,F2,
  s9 for natural_transformation of F2,F3,
  t for natural_transformation of G1,G2,
  t9 for natural_transformation of G2,G3,
  u for natural_transformation of H1,H2;

theorem Th47:
  A,B are_equivalent implies for F being Equivalence of A,B ex G
  being Equivalence of B,A st G*F ~= id A & F*G ~= id B
proof
  assume
A1: A,B are_equivalent;
  let F be Equivalence of A,B;
  consider G be Functor of B,A such that
A2: G*F ~= id A & F*G ~= id B by A1,Def11;
  G is Equivalence of B,A
  proof
    thus B,A are_equivalent by A1;
    take F;
    thus thesis by A2;
  end;
  hence thesis by A2;
end;
