reserve A,B,C for non empty set,
  f for Function of [:A,B:],C;
reserve K for non empty doubleLoopStr;
reserve V for non empty ModuleStr over K;
reserve W for non empty RightModStr over K;

theorem
  for K being strict non empty doubleLoopStr, W being non empty
  RightModStr over K holds opp(opp(W)) = the RightModStr of W
proof
  let K be strict non empty doubleLoopStr, W be non empty RightModStr over K;
  set V = opp(W);
A1: opp(opp(K)) = K by FUNCT_4:53;
A2: opp(the lmult of V) = opp(opp(the rmult of W)) by Th10
    .= the rmult of W by FUNCT_4:53;
  the addLoopStr of opp(V) = the addLoopStr of V by Th7
    .= the addLoopStr of W by Th9;
  hence thesis by A2,A1,Th8;
end;
