reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve H for BinOp of E;

theorem
  F is associative & F is having_a_unity & F is commutative &
  F is having_an_inverseOp & u = the_inverseOp_wrt F implies
  u.(F*(id D,u).(d1,d2)) =
    F*(u,id D).(d1,d2) & F*(id D,u).(d1,d2) = u.(F*(u,id D).(d1,d2))
proof
  assume that
A1: F is associative & F is having_a_unity and
A2: F is commutative and
A3: F is having_an_inverseOp & u = the_inverseOp_wrt F;
A4: u is_distributive_wrt F by A1,A2,A3,Th63;
  thus u.(F*(id D,u).(d1,d2)) = u.(F.(d1,u.d2)) by Th81
    .= F.(u.d1,u.(u.d2)) by A4
    .= F.(u.d1,d2) by A1,A3,Th62
    .= F*(u,id D).(d1,d2) by Th81;
  hence thesis by A1,A3,Th62;
end;
