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 Th31:
  <% <%0.N_Real,0.N_Real%> , <%1.N_Real,0.N_Real%> %> *
  <% <%0.N_Real,1.N_Real%> , <%0.N_Real,0.N_Real%> %>
  = <% <%0.N_Real,0.N_Real%> , <%0.N_Real,-1.N_Real%> %>
  proof
    ZJ*' = <%z*',-j%> by Def9;
    then ZJ*'*JZ = <%z*j-z*'*-j,z*z+j*'*-j%> by Def9;
    hence <%ZZ,ZM%> = <%ZZ*ZJ-ZZ*'*JZ,ZZ*ZZ+ZJ*'*JZ%>
    .= <%ZZ,JZ%> * <%ZJ,ZZ%> by Def9;
  end;
