theorem Th13:
  for c being Element of Cayley-Dickson(Cayley-Dickson(N))
  ex a1,b1,a2,b2 st c = <%<%a1,b1%>,<%a2,b2%>%>
  proof
    let c be Element of Cayley-Dickson(Cayley-Dickson(N));
    consider a,b being Element of Cayley-Dickson(N) such that
A1: c = <%a,b%> by Th12;
    consider a1,b1 being Element of N such that
A2: a = <%a1,b1%> by Th12;
    consider a2,b2 being Element of N such that
A3: b = <%a2,b2%> by Th12;
    take a1,b1,a2,b2;
    thus c = <%<%a1,b1%>,<%a2,b2%>%> by A1,A2,A3;
  end;
