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