reserve u,v,x,y,z,X,Y for set;
reserve r,s for Real;
reserve N for non empty ConjNormAlgStr;
reserve a,a1,a2,b,b1,b2 for Element of N;
reserve c,c1,c2 for Element of Cayley-Dickson(N);

theorem Th30:
  <% <%0.N_Real,1.N_Real%> , <%0.N_Real,0.N_Real%> %> *
  <% <%0.N_Real,0.N_Real%> , <%1.N_Real,0.N_Real%> %>
  = <% <%0.N_Real,0.N_Real%> , <%0.N_Real,1.N_Real%> %>
  proof
    thus <%ZZ,ZJ%> = <%ZJ*ZZ-JZ*'*ZZ,ZJ*JZ+ZZ*'*ZZ%> by Lm2
    .= <%ZJ,ZZ%> * <%ZZ,JZ%> by Def9;
  end;
