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 Th32:
  X is non empty implies v = [v`1,v`2] & length v >= 1
proof
  assume X is non empty; then
  reconsider X9=X as non empty set;
  reconsider v9=v as Element of free_magma X9;
  v9 in [:free_magma(X,v9`2),{v9`2}:] by Th25; then
  ex x,y being object st x in free_magma(X,v9`2) &
  y in {v9`2} & v9=[x,y] by ZFMISC_1:def 2;
  hence v = [v`1,v`2];
  reconsider v99=v9 as Element of free_magma_carrier X9;
  v99`2 > 0; then
  length v9 > 0 by Def18; then
  length v9+1 > 0+1 by XREAL_1:6;
  hence thesis by NAT_1:13;
end;
