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 Th33:
  <%<%<%0.N_Real,0.N_Real%>,<%1.N_Real,0.N_Real%>%>,
  <%<%0.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%>,<%0.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%>%>%>
  proof
A1: JZ*ZJ*' = <%j*z*'-(-j)*'*z,(-j)*j+z*z*'*'%> by Lm3,Def9
    .= <%z,-j%>;
A2: ZZJZ*ZJZZ-ZZZZ*'*ZZZZ = <%ZZ*ZJ-ZZ*'*JZ,ZZ*ZZ+JZ*ZJ*'%> by Def9
    .= <%ZZ,ZM%> by A1;
    ZZZZ*ZZJZ+ZZZZ*ZJZZ*' = ZZZZ;
    hence <%ZZJZ,ZZZZ%> * <%ZJZZ,ZZZZ%> = <%<%ZZ,ZM%>,ZZZZ%> by A2,Def9;
  end;
