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 Th32:
  <%<%<%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%>,<%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,1.N_Real%>%>,
  <%<%0.N_Real,0.N_Real%>,<%0.N_Real,0.N_Real%>%>%>
  proof
A1: ZJZZ*ZZJZ-ZZZZ*'*ZZZZ = <%ZJ*ZZ-JZ*'*ZZ,ZJ*JZ+ZZ*'*ZZ%> by Def9
    .= <%ZZ,ZJ*JZ%>
    .= ZZZJ by Lm2;
    ZZZZ*ZJZZ+ZZZZ*ZZJZ*' = ZZZZ;
    hence thesis by A1,Def9;
  end;
