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;

theorem Th26:
  for X being non empty set, v,w being Element of free_magma_carrier X
  holds [[[v`1,w`1],v`2],v`2+w`2] is Element of free_magma_carrier X
proof
  let X be non empty set;
  let v,w be Element of free_magma_carrier X;
  v in [:free_magma(X,v`2),{v`2}:] by Th25; then
  A1: v`1 in free_magma(X,v`2) by MCART_1:10;
  w in [:free_magma(X,w`2),{w`2}:] by Th25; then
  w`1 in free_magma(X,w`2) by MCART_1:10; then
  A2: [[v`1,w`1],v`2] in free_magma(X,v`2+w`2) by A1,Th22;
  A3: v`2 + w`2 in {v`2 + w`2} by TARSKI:def 1;
  set z = [[[v`1,w`1],v`2],v`2+w`2];
  A4: z`1 in free_magma(X,v`2+w`2) by A2;
  z`2 in {v`2 + w`2} by A3; then
  A5: z in [:free_magma(X,v`2+w`2),{v`2+w`2}:] by A4,ZFMISC_1:def 2;
  [:free_magma(X,v`2 + w`2),{v`2 + w`2}:] c= free_magma_carrier X by Lm1;
  hence thesis by A5;
end;
