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 Th31:
  X is non empty implies v*w = [[[v`1,w`1],v`2],length v + length w]
proof
  assume A1: X is non empty; then
  length v = v`2 & length w = w`2 by Def18;
  hence thesis by A1,Def16;
end;
