theorem Th43:
  F is having_a_unity implies F.:(C-->the_unity_wrt F,f) = f &
    F.:(f,C-->the_unity_wrt F) = f
proof
  set e = the_unity_wrt F;
  reconsider g = C-->e as Function of C,D;
  assume
A1: F is having_a_unity;
  now
    let c;
    thus (F.:(g,f)).c = F.(g.c,f.c) by FUNCOP_1:37
      .= F.(e,f.c)
      .= f.c by A1,SETWISEO:15;
  end;
  hence F.:(C-->e,f) = f by FUNCT_2:63;
  now
    let c;
    thus (F.:(f,g)).c = F.(f.c,g.c) by FUNCOP_1:37
      .= F.(f.c,e)
      .= f.c by A1,SETWISEO:15;
  end;
  hence thesis by FUNCT_2:63;
end;
