reserve x,y for set;
reserve C,C9,D,E for non empty set;
reserve c,c9,c1,c2,c3 for Element of C;
reserve B,B9,B1,B2 for Element of Fin C;
reserve A for Element of Fin C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve f,f9 for Function of C,D;
reserve g for Function of C9,D;
reserve H for BinOp of E;
reserve h for Function of D,E;
reserve i,j for Nat;
reserve s for Function;
reserve p,q for FinSequence of D;
reserve T1,T2 for Element of i-tuples_on D;

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"**"T1,F"**"
  T2) = F "**"(G.:(T1,T2))
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,Th33;
end;
