reserve M,N for non empty multMagma,
  f for Function of M, N;
reserve M for multMagma;
reserve N,K for multSubmagma of M;
reserve M,N for non empty multMagma,
  A for Subset of M,
  f,g for Function of M,N,
  X for stable Subset of M,
  Y for stable Subset of N;
reserve X for set;
reserve x,y,Y for set;
reserve n,m,p for Nat;
reserve v,v1,v2,w,w1,w2 for Element of free_magma X;

theorem
  length(v*w) = length v + length w
proof
  set vw = v*w;
  per cases;
  suppose A1: X is non empty; then
     v*w = [[[v`1,w`1],v`2],length v + length w] by Th31;
    hence length(v*w) = [[[v`1,w`1],v`2],length v + length w]`2 by A1,Def18
    .= length v + length w;
  end;
  suppose A2: X is empty;
    hence length(v*w) = 0 by Def18
    .= length v + 0 by A2,Def18
    .= length v + length w by A2,Def18;
  end;
end;
