theorem
  \Cup(M,A) \iff A\or(M*A)\or\Cup(M, (M\;M)*A) in H
  proof
A1: \Cup(M,A) \iff A\or\Cup(M, M*A) in H by Def43;
    \Cup(M,M*A) \iff (M*A)\or\Cup(M, M*(M*A)) in H by Def43;
    then
A2: \Cup(M,A) \iff A\or((M*A)\or\Cup(M, M*(M*A))) in H by A1,Th99;
    A\or(M*A)\or\Cup(M, M*(M*A))\iffA\or((M*A)\or\Cup(M, M*(M*A))) in H
    by Th76;
    then A\or((M*A)\or\Cup(M, M*(M*A)))\iffA\or(M*A)\or\Cup(M, M*(M*A)) in H
    by Th90;
    then
A3: \Cup(M,A) \iff A\or(M*A)\or\Cup(M, M*(M*A)) in H by A2,Th91;
    ((M\;M)*A)\iff(M*(M*A)) in H by Def43;
    then (M*(M*A)) \iff ((M\;M)*A) in H by Th90;
    then \Cup(M, M*(M*A))\iff\Cup(M, (M\;M)*A) in H by Th147;
    hence thesis by A3,Th99;
  end;
