reserve l, m, n for Nat,
  i,j,k for Instruction of SCM+FSA,
  I,J,K for Program of SCM+FSA;
reserve a,b for Int-Location,
  f for FinSeq-Location,
  s,s1,s2 for State of SCM+FSA;

theorem Th15:
  I ";" J ";" K = I ";" (J ";" K)
proof
A1: card stop I -' 1 = card I by COMPOS_1:71;
A2: card stop Directed I = card stop I by Lm2;
A3: card stop Directed(I ";" J) = card stop(I ";" J) by Lm2;
A4: card stop(I ";" J) -' 1 = card(I ";" J) by COMPOS_1:71;
A5: card stop Directed J = card stop J by Lm2;
   card stop J -' 1 = card J by COMPOS_1:71;
   then
A6: Reloc(J ";" K, card I)
   = Reloc(Directed J, card I) +* Reloc(Reloc(K, card J), card I)
               by COMPOS_1:42,A5
  .= Reloc(Directed J, card I) +* Reloc(K,card J + card I) by COMPOS_1:43;
  thus I ";" J ";" K
   = (I ";" Directed J) +* Reloc(K, card (I ";" J)) by Th14,A4,A3
  .= Directed I +* (Reloc(Directed J, card I) +*
         Reloc(K, card (I ";" J))) by FUNCT_4:14,A1,A2
  .= I ";" (J ";" K) by A6,Th11,A1,A2;
end;
