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;
