theorem
  F is commutative associative & F is having_a_unity & F is
having_an_inverseOp & G = F*(id D,the_inverseOp_wrt F) implies G.(F$$(B,f),F$$(
  B,f9)) = F $$(B,G.:(f,f9))
proof
  assume that
A1: F is commutative associative & F is having_a_unity and
A2: F is having_an_inverseOp & G = F*(id D,the_inverseOp_wrt F);
  set e = the_unity_wrt F;
  G.(e,e) = e & for d1,d2,d3,d4 holds F.(G.(d1,d2),G.(d3,d4))= G.(F.(d1,d3
  ),F. (d2,d4)) by A1,A2,FINSEQOP:86,89;
  hence thesis by A1,Th9;
end;
