theorem
  F is commutative associative & F is having_a_unity & F is
  having_an_inverseOp implies (the_inverseOp_wrt F).(F$$(B,f)) = F $$(B,(
  the_inverseOp_wrt F)*f)
proof
  assume that
A1: F is commutative associative & F is having_a_unity and
A2: F is having_an_inverseOp;
  set e = the_unity_wrt F, u = the_inverseOp_wrt F;
  u is_distributive_wrt F by A1,A2,FINSEQOP:63;
  then
A3: for d1,d2 holds u.(F.(d1,d2)) = F.(u.d1,u.d2);
  u.e = e by A1,A2,FINSEQOP:61;
  hence thesis by A1,A3,Th16;
end;
