theorem
  M*V in H & M*(M1*A) in H or M*\notV in H & M*(M2*A) in H implies
  if-then-else(M,M1,M2)*A in H
  proof
    assume M*V in H & M*(M1*A) in H or M*\notV in H & M*(M2*A) in H;
    then ((M*V)\and(M*(M1*A)) in H or (M*\notV)\and(M*(M2*A)) in H) &
    (M*V)\and(M*(M1*A))\imp((M*V)\and(M*(M1*A)))\or((M*\notV)\and(M*(M2*A)))
    in H &
    (M*\notV)\and(M*(M2*A))\imp((M*V)\and(M*(M1*A)))\or
    ((M*\notV)\and(M*(M2*A))) in H by Def38,Th35;
    then
A1: ((M*V)\and(M*(M1*A)))\or((M*\notV)\and(M*(M2*A))) in H by Def38;
    (if-then-else(M,M1,M2)*A) \iff
    ((M*V)\and(M*(M1*A)))\or((M*\notV)\and(M*(M2*A))) in H by Def43;
    then
    ((M*V)\and(M*(M1*A)))\or((M*\notV)\and(M*(M2*A)))\imp
    (if-then-else(M,M1,M2)*A) in H by Th43;
    hence thesis by Def38,A1;
  end;
